• The solution to Newton's GEXACT VALUE & FORMULA
  • The theory controlling planetary spinTHE MATHEMATICAL LAW
  • The pressure at the centre of a massEARTH'S CORE PRESSURE (calculation procedure)
  • Proof of the non-exitence of Dark MatterDOES NOT EXIST
  • The atom as Newton and Coulomb describe itNO NEED FOR A UNIFICATION THEORY
The solution to Newton's G1 The theory controlling planetary spin2 Pressure at the centre of the Earth3 Proof of the non-exitence of Dark Matter4 The atom as Newton describes it5
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Differentiation of Trigonometric Functions

The following table contains examples of differentiated trigonometric functions. Worked examples of many of those you see in this table are provided at the bottom of this page.

y = Sin(x)

dy/dx = Cos(x)

y = Sin(ax)

dy/dx = a.Cos(ax)

y = Sin(x/a)

dy/dx = 1/a .Cos(x/a)

y = Sin(a/x)

dy/dx = –a.Cos(a/x) / x²

y = Sin(x²)

dy/dx = 2x.Cos(x²)

y = Sin(ax²)

dy/dx = 2ax.Cos(ax²)

y = Sin(x²/a)

dy/dx = 2x.Cos(x²/a) / a

y = Sin(a/x²)

dy/dx = –2a.Cos(a/x²) / x³

y = Sin²(x)

dy/dx = Sin(2x)

y = Sin²(ax)

dy/dx = a.Sin(2ax)

y = Sin²(x/a)

dy/dx = Sin(2x/a)/a

y = Sin²(a/x)

dy/dx = –a.Sin(a/x)/x²

y = Sin²(x²)

dy/dx = 2x.Sin(2x²)

y = Sin²(ax²)

dy/dx = 2ax.Sin(2ax²)

y = Sin²(x²/a)

dy/dx = 2x.Sin(2x²/a)/a

y = Sin²(a/x²)

dy/dx = –2a.Sin(2a/x²) / x³

y = Cos(x)

dy/dx = –Sin(x)

y = Cos(ax)

dy/dx = –a .Sin(ax)

y = Cos(x/a)

dy/dx = –1 / a.Sin(x/a)

y = Cos(a/x)

dy/dx = a.Sin(a/x) / x²

y = Cos(x²)

dy/dx = –2x.Sin(x²)

y = Cos(ax²)

dy/dx = –2ax.Sin(ax²)

y = Cos(x²/a)

dy/dx = –2x.Sin(x²/a) / a

y = Cos(a/x²)

dy/dx = 2a.Sin(a/x²) / x³

y = Cos²(x)

dy/dx = –2.Sin(2x)

y = Cos²(ax)

dy/dx = –a.Sin(2ax)

y = Cos²(x/a)

dy/dx = –Sin(2x/a) / a

y = Cos²(a/x)

dy/dx = Sin(a/x) / x²

y = Cos²(x²)

dy/dx = –2x.Sin(2x²)

y = Cos²(ax²)

dy/dx = –2ax.Sin(2ax²)

y = Cos²(x²/a)

dy/dx = –2x.Sin(2x²/a) / a

y = Cos²(a/x²)

dy/dx = 2a.Sin(2a/x²) / x³

y = Tan(x)

dy/dx = 1 / Cos²(x)

y = Tan(ax)

dy/dx = a / Cos²(ax)

y = Tan(x/a)

dy/dx = 1 / a.Cos²(x/a)

y = Tan(a/x)

dy/dx = –a / x².Cos²(a/x)

y = Tan(x²)

dy/dx = 2x / Cos²(x²)

y = Tan(ax²)

dy/dx = 2ax / Cos²(ax²)

y = Tan(x²/a)

dy/dx = 2x / a.Cos²(x²/a)

y = Tan(a/x²)

dy/dx = –2a / x³.Cos²(a/x²)

y = Tan²(x)

dy/dx = 2.Tan(x) / Cos²(x)

y = Tan²(ax)

dy/dx = 2a.Tan(ax) / Cos²(ax)

y = Tan²(x/a)

dy/dx = 2.tan(x/a) / a.Cos²(x/a)

y = Tan²(a/x)

dy/dx = –2a.Tan(a/x) / x².Cos²(a/x)

y = Tan²(x²)

dy/dx = 4x.Tan(x²) / Cos²(x²)

y = Tan²(ax²)

dy/dx = 4ax.Tan(ax²) / Cos²(ax²)

y = Tan²(x²/a)

dy/dx = 4x.Tan(x²/a) / a.Cos²(x²/a)

y = Tan²(a/x²)

dy/dx = –4a.Tan(a/x²) / x³.Cos²(a/x²)

y = Csc(x)

dy/dx = – 1 / Sin(x).Tan(x)

y = Csc(ax)

dy/dx = –a / Sin(ax).Tan(ax)

y = Csc(x/a)

dy/dx = –1 / a.Sin(x/a).Tan(x/a)

y = Csc(a/x)

dy/dx = a / x².Sin(a/x).Tan(a/x)

y = Csc(x²)

dy/dx = –2x / Sin(x²).Tan(x²)

y = Csc(ax²)

dy/dx = –2ax / Sin(ax²).Tan(ax²)

y = Csc(x²/a)

dy/dx = –2x / a.Sin(x²/a).Tan(x²/a)

y = Csc(a/x²)

dy/dx = 2a / x³.Sin(a/x²).Tan(a/x²)

y = Csc²(x)

dy/dx = –2 / Tan(x).Sin²(x)

y = Csc²(ax)

dy/dx = –2a / Sin²(ax).Tan²(ax)

y = Csc²(x/a)

dy/dx = –2 / a.Sin²(x/a).Tan(x/a)

y = Csc²(a/x)

dy/dx = 2a / x².Sin²(a/x).Tan(a/x)

y = Csc²(x²)

dy/dx = –4x / Sin²(x²).Tan(x²)

y = Csc²(ax²)

dy/dx = –4x / Tan(x²)/Sin²(x²)

y = Csc²(x²/a)

dy/dx = –4x / a.Tan(x²/a).Sin²(x²/a)

y = Csc²(a/x²)

dy/dx = 4a / x³.Tan(ax²).Sin²(ax²)

y = Sec(x)

dy/dx = Tan(x) / Cos(x)

y = Sec(ax)

dy/dx = a.Tan(ax) / Cos(ax)

y = Sec(x/a)

dy/dx = Tan(x/a) / a.Cos(x/a)

y = Sec(a/x)

dy/dx = –a.Tan(a/x) / x².Cos(a/x)

y = Sec(x²)

dy/dx = 2x.Tan(x²) / Cos(x²)

y = Sec(ax²)

dy/dx = 2ax.Tan(ax²) / Cos(ax²)

y = Sec(x²/a)

dy/dx = 2x.Tan(x²/a) / a.Cos(x²/a)

y = Sec(a/x²)

dy/dx = –2a.Tan(a/x²) / x³.Cos(a/x²)

y = Sec²(x)

dy/dx = 2.Tan(x) / Cos²(x)

y = Sec²(ax)

dy/dx = 2a.Tan(ax) / Cos²(ax)

y = Sec²(x/a)

dy/dx = 2.Tan(x/a) / Cos²(x/a)

y = Sec²(a/x)

dy/dx = –2a . Tan(a/x) / x².Cos²(a/x)

y = Sec²(x²)

dy/dx = 4x.Tan(x²) / Cos²(x²)

y = Sec²(ax²)

dy/dx = 4ax.Tan(ax²) / Cos²(ax²)

y = Sec²(x²/a)

dy/dx = 4x.Tan(x²/a) / a.Cos²(x²/a)

y = Sec²(a/x²)

dy/dx = –4a.Tan(ax²) / x³.Cos²(ax²)

y = Cot(x)

dy/dx = –1 / Sin²(x)

y = Cot(ax)

dy/dx = –a / Sin²(ax)

y = Cot(x/a)

dy/dx = –1 / a.Sin²(x/a)

y = Cot(a/x)

dy/dx = a / x².Sin²(a/x)

y = Cot(x²)

dy/dx = –2x / Sin²(x²)

y = Cot(ax²)

dy/dx = –2ax / Sin²(ax²)

y = Cot(x²/a)

dy/dx = –2x / a.Sin²(x²/a)

y = Cot(a/x²)

dy/dx = 2a / x³.Sin²(a/x²)

y = Cot²(x)

dy/dx = –2 / Tan(x).Sin²(x)

y = Cot²(ax)

dy/dx = –2a / Tan(ax).Sin²(ax)

y = Cot²(x/a)

dy/dx = –2 / a.Tan(x/a).Sin²(x/a)

y = Cot²(a/x)

dy/dx = 2a / x².Tan(a/x).Sin²(a/x)

y = Cot²(x²)

dy/dx = –4x / Tan(x²).Sin²(x²)

y = Cot²(ax²)

dy/dx = –4ax / Tan(ax²).Sin²(ax²)

y = Cot²(x²/a)

dy/dx = –4x / a.Tan(x²/a).Sin²(x²/a)

y = Cot²(a/x²)

dy/dx = 4a / x³.Tan(a/x²).Sin²(a/x²)

y = Asin(x)

dy/dx = 1 / (1 – x²)½

y = Acos(x)

dy/dx = -1 / (1 – x²)½

y = Atan(x)

dy/dx = 1 / (1 + x²)

y = Acot(x)

dy/dx = -1 / (1 + x²)

y = Asec(x)

dy/dx = 1 / x(x² – 1)

y = Acsc(x)

dy/dx = -1 / x(x² – 1)

y = Asin(x/a)

dy/dx = 1 / (a² – x²)½

y = Acos(x/a)

dy/dx = -1 / (a² – x²)½

y = Atan(x/a)

dy/dx = a / (a² + x²)

y = Acot(x/a)

dy/dx = -a / (a² + x²)

y = Asec(x/a)

dy/dx = a / x(x² – a²)

y = Acsc(x/a)

dy/dx = -a / x(x² – a²)

y = Sinh(x)

dy/dx = Cosh(x)

y = Cosh(x)

dy/dx = Sinh(x)

y = Tanh(x)

dy/dx = Sech²(x)

y = Csch(x)

dy/dx = -Csch(x).Coth(x)

y = Sech(x)

dy/dx = -Sech(x).Tanh(x)

y = Coth(x)

dy/dx = -Csch²(x)

y = Asinh(x)

dy/dx =1 / (x² + 1)½

y = Acosh(x)

dy/dx =1 / (x² – 1)½

y = Atanh(x)

dy/dx =1 / (1 – x²)

y = Asech(x)

dy/dx = –1 / x(1 – x²)½

y = Acsch(x)

dy/dx = –1 / x(1 + x²)½

y = Acoth(x)

dy/dx = –1 / (x² – 1)

y = Asinh(x/a)

dy/dx = 1 / (x² + a²)½

y = Acosh(x/a)

dy/dx = 1 / (x² – a²)½

y = Atanh(x/a)

dy/dx = a / (a² – x²)

Worked Examples

Sin & Cos

y = Sin(a/x)
y = Sin(u) > dy/du = Cos(u)
u = a/x > du/dx = –a/x²
dy/dx = dy/du . du/dx
dy/dx = Cos(u) . –a/x²
dy/dx = Cos(a/x) . –a/x²
dy/dx = –a.Cos(a/x) / x²

y = Cos(a/x)
y = Cos(u) > dy/du = –Sin(u)
u = a/x > du/dx = –a/x²
dy/dx = dy/du . du/dx
dy/dx = –Sin(u) . –a/x²
dy/dx = –Sin(a/x) . –a/x²
dy/dx = a.Sin(a/x) / x²

y = Sin(x²)
y = Sin(u) > dy/du = Cos(u)
u = x² > du/dx = 2x
dy/dx = dy/du . du/dx
dy/dx = Cos(u) . 2x
dy/dx = Cos(x²) . 2x
dy/dx = 2x.Cos(x²)

y = Cos(x²)
y = Cos(u) > dy/du = –Sin(u)
u = x² > du/dx = 2x
dy/dx = dy/du . du/dx
dy/dx = –Sin(u) . 2x
dy/dx = –Sin(x²) . 2x
dy/dx = –2x.Sin(x²)

y = Sin(ax²)
y = Sin(u) > dy/du = Cos(u)
u = ax² > du/dx = 2ax
dy/dx = dy/du . du/dx
dy/dx = Cos(u) . 2ax
dy/dx = Cos(ax²) . 2ax
dy/dx = 2ax.Cos(ax²)

y = Cos(ax²)
y = Cos(u) > dy/du = –Sin(u)
u = ax² > du/dx = 2ax
dy/dx = dy/du . du/dx
dy/dx = –Sin(u) . 2ax
dy/dx = –Sin(ax²) . 2ax
dy/dx = –2ax.Sin(ax²)

y = Sin(x²/a)
y = Sin(u) > dy/du = Cos(u)
u = x²/a > du/dv = 2x/a
dy/dx = dy/du . du/dx
dy/dx = Cos(u) . 2x/a
dy/dx = Cos(x²/a) . 2x/a
dy/dx = 2x.Cos(x²/a) / a

y = Cos(x²/a)
y = Cos(u) > dy/du = –Sin(u)
u = x²/a > du/dv = 2x/a
dy/dx = dy/du . du/dx
dy/dx = –Sin(u) . 2x/a
dy/dx = –Sin(x²/a) . 2x/a
dy/dx = –2x.Sin(x²/a) / a

y = Sin(a/x²)
y = Sin(u) > dy/du = Cos(u)
u = a/x² > du/dv = –2a/x³
dy/dx = dy/du . du/dx
dy/dx = Cos(u) . –2a/x³
dy/dx = Cos(a/x²) . –2a/x³
dy/dx = –2a.Cos(a/x²) / x³

y = Cos(a/x²)
y = Cos(u) > dy/du = –Sin(u)
u = a/x² > du/dv = –2a/x³
dy/dx = dy/du . du/dx
dy/dx = –Sin(u) . –2a/x³
dy/dx = –Sin(a/x²) . –2a/x³
dy/dx = 2a.Sin(a/x²) / x³

y = Sin²(x) = [Sin(x)]²
y = u² > dy/du = 2u
u = Sin(x) > du/dv = Cos(x)
dy/dx = dy/du . du/dx
dy/dx = 2u . Cos(x)
dy/dx = 2.Sin(x) . Cos(x)
dy/dx = Sin(2x)

y = Cos²(x) = [Cos(x)]²
y = u² > dy/du = 2u
u = Cos(x) > du/dv = –Sin(x)
dy/dx = dy/du . du/dx
dy/dx = 2u . –Sin(x)
dy/dx = 2.Cos(x) . –Sin(x)
dy/dx = –2.Sin(2x)

y = Sin²(ax) = [Sin(ax)]²
y = u² > dy/du = 2u
u = Sin(v) > du/dv = Cos(v)
v = ax > dv/dx = a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Cos(v) . a
dy/dx = 2.Sin(v) . Cos(v) . a
dy/dx = 2.Sin(ax) . Cos(ax) . a
= 2a . Sin(ax) . Cos(ax)
dy/dx = a.Sin(2ax)

y = Cos²(ax) = [Cos(ax)]²
y = u² > dy/du = 2u
u = Cos(v) > du/dv = –Sin(v)
v = ax > dv/dx = a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Sin(v) . a
dy/dx = 2.Cos(v) . –Sin(v) . a
dy/dx = 2.Cos(ax) . –Sin(ax) . a
= 2a . Cos(ax) . –Sin(ax)
dy/dx = –a.Sin(2ax)

y = Sin²(x/a) = [Sin(x/a)]²
y = u² > dy/du = 2u
u = Sin(v) > du/dv = Cos(v)
v = x/a > dv/dx = 1/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Cos(v) / a
dy/dx = 2.Sin(v) . Cos(v) / a
dy/dx = 2.Sin(x/a) . Cos(x/a) / a
dy/dx = Sin(2x/a) / a

y = Cos²(x/a) = [Cos(x/a)]²
y = u² > dy/du = 2u
u = Cos(v) > du/dv = –Sin(v)
v = x/a > dv/dx = 1/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Sin(v) / a
dy/dx = 2.Cos(v) . –Sin(v) / a
dy/dx = 2.Cos(x/a) . –Sin(x/a) / a
dy/dx = –Sin(2x/a) / a

y = Sin²(a/x) = [Sin(a/x)]²
y = u² > dy/du = 2u
u = Sin(v) > du/dv = Cos(v)
v = a/x > dv/dx = –a/x²
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Cos(v) . –a/x²
dy/dx = 2.Sin(v) . Cos(v) . –a/x²
dy/dx = 2.Sin(a/x) . Cos(a/x) . –a/x²
= –2a . Sin(a/x) . Cos(a/x) / x²
dy/dx = –a.Sin(a/x) / x²

y = Cos²(a/x) = [Cos(a/x)]²
y = u² > dy/du = 2u
u = Cos(v) > du/dv = –Sin(v)
v = a/x > dv/dx = –a/x²
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Sin(v) . –a/x²
dy/dx = 2.Cos(v) . –Sin(v) . –a/x²
dy/dx = 2.Cos(a/x) . –Sin(a/x) . –a/x²
= 2a . Cos(a/x) . Sin(a/x) / x²
dy/dx = a.Sin(a/x) / x²

y = Sin²(x²) = [Sin(x²)]²
y = u² > dy/du = 2u
u = Sin(v) > du/dv = Cos(v)
v = x² > dv/dx = 2x
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Cos(v) . 2x
dy/dx = 2.Sin(v) . Cos(v) . 2x
dy/dx = 2.Sin(x²) . Cos(x²) . 2x
= 4x . Sin(x²) . Cos(x²)
dy/dx = 2x.Sin(2x²)

y = Cos²(x²) = [Cos(x²)]²
y = u² > dy/du = 2u
u = Cos(v) > du/dv = –Sin(v)
v = x² > dv/dx = 2x
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Sin(v) . 2x
dy/dx = 2.Cos(v) . –Sin(v) . 2x
dy/dx = 2.Cos(x²) . –Sin(x²) . 2x
= –4x . Cos(x²) . Sin(x²)
dy/dx = –2x.Sin(2x²)

y = Sin²(ax²) = [Sin(ax²)]²
y = u² > dy/du = 2u
u = Sin(v) > du/dv = Cos(v)
v = ax² > dv/dx = 2ax
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Cos(v) . 2ax
dy/dx = 2.Sin(v) . Cos(v) . 2ax
dy/dx = 2.Sin(ax²) . Cos(ax²) . 2ax
= 4ax . Sin(ax²) . Cos(ax²)
dy/dx = 2ax.Sin(2ax²)

y = Cos²(ax²) = [Cos(ax²)]²
y = u² > dy/du = 2u
u = Cos(v) > du/dv = –Sin(v)
v = ax² > dv/dx = 2ax
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Sin(v) . 2ax
dy/dx = 2.Cos(v) . –Sin(v) . 2ax
dy/dx = 2.Cos(ax²) . –Sin(ax²) . 2ax
= 4ax . Cos(ax²) . –Sin(ax²)
dy/dx = –2ax.Sin(2ax²)

y = Sin²(x²/a) = [Sin(x²/a)]²
y = u² > dy/du = 2u
u = Sin(v) > du/dv = Cos(v)
v = x²/a dv/dx = 2x/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Cos(v) . 2x/a
dy/dx = 2.Sin(v) . Cos(v) . 2x/a
dy/dx = 2.Sin(x²/a) . Cos(x²/a) . 2x/a
= 4x . Sin(x²/a) . Cos(x²/a) / a
dy/dx = 2x.Sin(2x²/a) / a

y = Cos²(x²/a) = [Cos(x²/a)]²
y = u² > dy/du = 2u
u = Cos(v) > du/dv = –Sin(v)
v = x²/a dv/dx = 2x/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Sin(v) . 2x/a
dy/dx = 2.Cos(v) . –Sin(v) . 2x/a
dy/dx = 2.Cos(x²/a) . –Sin(x²/a) . 2x/a
= 4x . Cos(x²/a) . –Sin(x²/a) / a
dy/dx = –2x.Sin(2x²/a) / a

y = Sin²(a/x²) = [Sin(a/x²)]²
y = u² > dy/du = 2u
u = Sin(v) > du/dv = Cos(v)
v = a/x² dv/dx = –2a/x³
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Cos(v) . –2a/x³
dy/dx = 2.Sin(v) . Cos(v) . –2a/x³
dy/dx = 2.Sin(a/x²) . Cos(a/x²) . –2a/x³
= –4a . Sin(a/x²) . Cos(a/x²) / x³
dy/dx = –2a.Sin(2a/x²)/x³

y = Cos²(a/x²) = [Cos(a/x²)]²
y = u² > dy/du = 2u
u = Cos(v) > du/dv = –Sin(v)
v = a/x² > dv/dx = –2a/x³
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Sin(v) . –2a/x³
dy/dx = 2.Cos(v) . –Sin(v) . –2a/x³
dy/dx = 2.Cos(a/x²) . –Sin(a/x²) . –2a/x³
= 4a . Cos(a/x²) . Sin(a/x²) / x³
dy/dx = 2a.Sin(2a/x²)/x³

Csc & Sec

y = Csc(a/x)
y = Csc(u) > dy/du = –Csc(u).Cot(u)
u = a/x > du/dv = –a/x²
dy/dx = dy/du . du/dx
dy/dx = –Csc(u).Cot(u) . –a/x²
dy/dx = –Csc(a/x).Cot(a/x) . –a/x²
dy/dx = a / x².Sin(a/x).Tan(a/x)

y = Sec(a/x)
y = Sec(u) > dy/du = Sec(u).Tan(u)
u = a/x > du/dv = –a/x²
dy/dx = dy/du . du/dx
dy/dx = Sec(u).Tan(u) . –a/x²
dy/dx = Sec(a/x).Tan(a/x) . –a/x²
dy/dx = –a.Tan(a/x) / x².Cos(a/x)

y = Csc(x²)
y = Csc(u) > dy/du = –Csc(u).Cot(u)
u = x² > du/dv = 2x
dy/dx = dy/du . du/dx
dy/dx = –Csc(u).Cot(u) . 2x
dy/dx = –Csc(x²).Cot(x²) . 2x
dy/dx = –2x / Sin(x²).Tan(x²)

y = Sec(x²)
y = Sec(u) > dy/du = Sec(u).Tan(u)
u = x² > du/dv = 2x
dy/dx = dy/du . du/dx
dy/dx = Sec(u).Tan(u) . 2x
dy/dx = Sec(x²).Tan(x²) . 2x
dy/dx = 2x.Tan(x²) / Cos(x²)

y = Csc(ax²)
y = Csc(u) > dy/du = –Csc(u).Cot(u)
u = ax² > du/dv = 2ax
dy/dx = dy/du . du/dx
dy/dx = –Csc(u).Cot(u) . 2ax
dy/dx = –Csc(ax²).Cot(ax²) . 2ax
dy/dx = –2ax / Sin(ax²).Tan(ax²)

y = Sec(ax²)
y = Sec(u) > dy/du = Sec(u).Tan(u)
u = ax² > du/dv = 2ax
dy/dx = dy/du . du/dx
dy/dx = Sec(u).Tan(u) . 2ax
dy/dx = Sec(ax²).Tan(ax²) . 2ax
dy/dx = 2ax.Tan(ax²) / Cos(ax²)

y = Csc(x²/a)
y = Csc(u) > dy/du = –Csc(u).Cot(u)
u = x²/a > du/dv = 2x/a
dy/dx = dy/du . du/dx
dy/dx = –Csc(u).Cot(u) . 2x/a
dy/dx = –Csc(x²/a).Cot(x²/a) . 2x/a
dy/dx = –2x / a.Sin(x²/a).Tan(x²/a)

y = Sec(x²/a)
y = Sec(u) > dy/du = Sec(u).Tan(u)
u = x²/a > du/dv = 2x/a
dy/dx = dy/du . du/dx
dy/dx = Sec(u).Tan(u) . 2x/a
dy/dx = Sec(x²/a).Tan(x²/a) . 2x/a
dy/dx = 2x.Tan(x²/a) / a.Cos(x²/a)

y = Csc(a/x²)
y = Csc(u) > dy/du = –Csc(u).Cot(u)
u = a/x² > du/dv = –2a/x³
dy/dx = dy/du . du/dx
dy/dx = –Csc(u).Cot(u) . –2a/x³
dy/dx = –Csc(a/x²).Cot(a/x²) . –2a/x³
dy/dx = 2a / x³.Sin(a/x²).Tan(a/x²)

y = Sec(a/x²)
y = Sec(u) > dy/du = Sec(u).Tan(u)
u = a/x² > du/dv = –2a/x³
dy/dx = dy/du . du/dx
dy/dx = Sec(u).Tan(u) . –2a/x³
dy/dx = Sec(a/x²).Tan(a/x²) . –2a/x³
dy/dx = –2a.Tan(a/x²) / x³.Cos(a/x²)

y = Csc²(x) = [Csc(x)]²
y = u² > dy/du = 2u
u = Csc(x) > du/dv = –Csc(x).Cot(x)
dy/dx = dy/du . du/dx
dy/dx = 2u . –Csc(x).Cot(x)
dy/dx = 2.Csc(x) . –Csc(x).Cot(x)
= –2.Csc²(x) . Cot(x)
dy/dx = –2 / Tan(x).Sin²(x)

y = Sec²(x) = [Sec(x)]²
y = u² > dy/du = 2u
u = Sec(x) > du/dv = Sec(x).Tan(x)
dy/dx = dy/du . du/dx
dy/dx = 2u . Sec(x).Tan(x)
dy/dx = 2.Sec(x) . Sec(x).Tan(x)
= 2.Sec²(x) . Tan(x)
dy/dx = 2.Tan(x) / Cos²(x)

y = Csc²(ax) = [Csc(ax)]²
y = u² > dy/du = 2u
u = Csc(v) > du/dv = –Csc(v).Cot(v)
v = ax > dv/dx = a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc(v).Cot(v) . a
dy/dx = 2.Csc(v) . –Csc(v).Cot(v) . a
dy/dx = 2 . Csc(ax) . –Csc(ax).Cot(ax) . a
= –2a . Csc²(ax) . Cot(ax) . a
dy/dx = –2a / Sin²(ax).Tan²(ax)

y = Sec²(ax) = [Sec(ax)]²
y = u² > dy/du = 2u
u = Sec(v) > du/dv = Sec(v).Tan(v)
v = ax > dv/dx = a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec(v).Tan(v) . a
dy/dx = 2.Sec(v) . Sec(v).Tan(v) . a
dy/dx = 2.Sec(ax) . Sec(ax).Tan(ax) . a
= 2a . Sec²(ax) . Tan(ax)
dy/dx = 2a.Tan(ax) / Cos²(ax)

y = Csc²(x/a) = [Csc(x/a)]²
y = u² > dy/du = 2u
u = Csc(v) > du/dv = –Csc(v).Cot(v)
v = x/a > dv/dx = 1/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc(v).Cot(v) / a
dy/dx = 2.Csc(v) . –Csc(v).Cot(v) / a
dy/dx = 2.Csc(x/a) . –Csc(x/a).Cot(x/a) / a
= –2 . Csc²(x/a).Cot(x/a) / a
dy/dx = –2 / a.Sin²(x/a).Tan(x/a)

y = Sec²(x/a) = [Sec(x/a)]²
y = u² > dy/du = 2u
u = Sec(v) > du/dv = Sec(v).Tan(v)
v = x/a > dv/dx = 1/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec(v).Tan(v) / a
dy/dx = 2.Sec(v) . Sec(v).Tan(v) / a
dy/dx = 2.Sec(x/a) . Sec(x/a).Tan(x/a) / a
= 2 . Sec²(x/a) . Tan(x/a) / a
dy/dx = 2.Tan(x/a) / a.Cos²(x/a)

y = Csc²(a/x) = [Csc(a/x)]²
y = u² > dy/du = 2u
u = Csc(v) > du/dv = –Csc(v).Cot(v)
v = a/x > dv/dx = –a/x²
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc(v).Cot(v) . –a/x²
dy/dx = 2.Csc(v) . –Csc(v).Cot(v) . –a/x²
dy/dx = 2.Csc(a/x) . –Csc(a/x).Cot(a/x) . –a/x²
= 2a . Csc²(a/x).Cot(a/x) / x²
dy/dx = 2a / x².Sin²(a/x).Tan(a/x)

y = Sec²(a/x) = [Sec(a/x)]²
y = u² > dy/du = 2u
u = Sec(v) > du/dv = Sec(v).Tan(v)
v = a/x > dv/dx = –a/x²
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec(v).Tan(v) . –a/x²
dy/dx = 2.Sec(v) . Sec(v).Tan(v) . –a/x²
dy/dx = 2.Sec(a/x) . Sec(a/x).Tan(a/x) . –a/x²
= –2a . Sec²(a/x) . Tan(a/x) / x²
dy/dx = –2a . Tan(a/x) / x².Cos²(a/x)

y = Csc²(x²) = [Csc(x²)]²
y = u² > dy/du = 2u
u = Csc(v) > du/dv = –Csc(v).Cot(v)
v = x² > dv/dx = 2x
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc(v).Cot(v) . 2x
dy/dx = 2.Csc(v) . –Csc(v).Cot(v) . 2x
dy/dx = 2.Csc(x²) . –Csc( x²).Cot( x²) . 2x
= –4x . Csc²(x²) . Cot(x²)
dy/dx = –4x / Sin²(x²).Tan(x²)

y = Sec²(x²) = [Sec(x²)]²
y = u² > dy/du = 2u
u = Sec(v) > du/dv = Sec(v).Tan(v)
v = x² > dv/dx = 2x
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec(v).Tan(v) . 2x
dy/dx = 2.Sec(v) . Sec(v).Tan(v) . 2x
dy/dx = 2.Sec(x²) . Sec(x²).Tan(x²) . 2x
= 4x . Sec²(x²) . Tan(x²)
dy/dx = 4x.Tan(x²) / Cos²(x²)

y = Csc²(ax²) = [Csc(ax²)]²
y = u² > dy/du = 2u
u = Csc(v) > du/dv = –Csc(v).Cot(v)
v = ax² > dv/dx = 2ax
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc(v).Cot(v) . 2ax
dy/dx = 2.Csc(v) . –Csc(v).Cot(v) . 2ax
dy/dx = 2.Csc(ax²) . –Csc(ax²).Cot(ax²) . 2ax
= –4ax . Csc²(ax²) . Cot(ax²)
dy/dx = –4ax / Sin²(ax²).Tan(ax²)

y = Sec²(ax²) = [Sec(ax²)]²
y = u² > dy/du = 2u
u = Sec(v) > du/dv = Sec(v).Tan(v)
v = ax² > dv/dx = 2ax
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec(v).Tan(v) . 2ax
dy/dx = 2.Sec(v) . Sec(v).Tan(v) . 2ax
dy/dx = 2.Sec(ax²) . Sec(ax²).Tan(ax²) . 2ax
= 4ax . Sec²(ax²) . Tan(ax²)
dy/dx = 4ax.Tan(ax²) / Cos²(ax²)

y = Csc²(x²/a) = [Csc(x²/a)]²
y = u² > dy/du = 2u
u = Csc(v) > du/dv = –Csc(v).Cot(v)
v = x²/a > dv/dx = 2x/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc(v).Cot(v) . 2x/a
dy/dx = 2.Csc(v) . –Csc(v).Cot(v) . 2x/a
dy/dx = 2.Csc(x²/a) . –Csc(x²/a).Cot(x²/a) . 2x/a
= –4x.Csc²(x²/a) . Cot(x²/a) / a
dy/dx = –4x / a.Tan(x²/a).Sin²(x²/a)

y = Sec²(x²/a) = [Sec(x²/a)]²
y = u² > dy/du = 2u
u = Sec(v) > du/dv = Sec(v).Tan(v)
v = x²/a > dv/dx = 2x/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec(v).Tan(v) . 2x/a
dy/dx = 2.Sec(v) . Sec(v).Tan(v) . 2x/a
dy/dx = 2.Sec(x²/a) . Sec(x²).Tan(x²/a) . 2x/a
= 4x.Sec²(x²/a) . Tan(x²/a) / a
dy/dx = 4x.Tan(x²/a) / a.Cos²(x²/a)

y = Csc²(a/x²) = [Csc(a/x²)]²
y = u² > dy/du = 2u
u = Csc(v) > du/dv = –Csc(v).Cot(v)
v = a/x² > dv/dx = –2a/x³
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc(v).Cot(v) . –2a/x³
dy/dx = 2.Csc(v) . –Csc(v).Cot(v) . –2a/x³
dy/dx = 2.Csc(ax²) . –Csc(ax²).Cot( ax²) . –2a/x³
= 4a.Csc²(ax²) . Cot(ax²) / x³
dy/dx = 4a / x³.Tan(ax²).Sin²(ax²)

y = Sec²(a/x²) = [Sec(a/x²)]²
y = u² > dy/du = 2u
u = Sec(v) > du/dv = Sec(v).Tan(v)
v = a/x² > dv/dx = –2a/x³
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec(v).Tan(v) . –2a/x³
dy/dx = 2.Sec(v) . Sec(v).Tan(v) . –2a/x³
dy/dx = 2.Sec(ax²) . Sec(ax²).Tan(ax²) . –2a/x³
= –4a.Sec²(ax²) . Tan(ax²) / x³
dy/dx = –4a.Tan(ax²) / x³.Cos²(ax²)

Tan & Cot

y = Tan(a/x)
y = Tan(u) > dy/du = Sec²(u)
u = a/x > du/dv = –a/x²
dy/dx = dy/du . du/dx
dy/dx = Sec²(u) . –a/x²
dy/dx = Sec²(a/x) . –a/x²
dy/dx = –a / x².Cos²(a/x)

y = Cot(a/x)
y = Cot(u) > dy/du = –Csc²(u)
u = a/x > du/dv = –a/x²
dy/dx = dy/du . du/dx
dy/dx = –Csc²(u) . –a/x²
dy/dx = –Csc²(a/x) . –a/x²
dy/dx = a / x².Sin²(a/x)

y = Tan(x²)
y = Tan(u) > dy/du = Sec²(u)
u = x² > du/dv = 2x
dy/dx = dy/du . du/dx
dy/dx = Sec²(u) . 2x
dy/dx = Sec²(x²) . 2x
dy/dx = 2x / Cos²(x²)

y = Cot(x²)
y = Cot(u) > dy/du = –Csc²(u)
u = x² > du/dv = 2x
dy/dx = dy/du . du/dx
dy/dx = –Csc²(u) . 2x
dy/dx = –Csc²(x²) . 2x
dy/dx = –2x / Sin²(x²)

y = Tan(ax²)
y = Tan(u) > dy/du = Sec²(u)
u = ax² > du/dv = 2ax
dy/dx = dy/du . du/dx
dy/dx = Sec²(u) . 2ax
dy/dx = Sec²(ax²) . 2ax
dy/dx = 2ax / Cos²(ax²)

y = Cot(ax²)
y = Cot(u) > dy/du = –Csc²(u)
u = ax² > du/dv = 2ax
dy/dx = dy/du . du/dx
dy/dx = –Csc²(u) . 2ax
dy/dx = –Csc²(ax²) . 2ax
dy/dx = –2ax / Sin²(ax²)

y = Tan(x²/a)
y = Tan(u) > dy/du = Sec²(u)
u = x²/a > du/dv = 2x/a
dy/dx = dy/du . du/dx
dy/dx = Sec²(u) . 2x/a
dy/dx = Sec²(x²/a) . 2x/a
dy/dx = 2x / a.Cos²(x²/a)

y = Cot(x²/a)
y = Cot(u) > dy/du = –Csc²(u)
u = x²/a > du/dv = 2x/a
dy/dx = dy/du . du/dx
dy/dx = –Csc²(u) . 2x/a
dy/dx = –Csc²(x²/a) . 2x/a
dy/dx = –2x / a.Sin²(x²/a)

y = Tan(a/x²)
y = Tan(u) > dy/du = Sec²(u)
u = a/x² > du/dv = –2a/x³
dy/dx = dy/du . du/dx
dy/dx = Sec²(u) . –2a/x³
dy/dx = Sec²(a/x²) . –2a/x³
dy/dx = –2a / x³.Cos²(a/x²)

y = Cot(a/x²)
y = Cot(u) > dy/du = –Csc²(u)
u = a/x² > du/dv = –2a/x³
dy/dx = dy/du . du/dx
dy/dx = –Csc²(u) . –2a/x³
dy/dx = –Csc²(a/x²) . –2a/x³
dy/dx = 2a / x³.Sin²(a/x²)

y = Tan²(x) = [Tan(x)]²
y = u² > dy/du = 2u
u = Tan(x) > du/dv = Sec²(x)
dy/dx = dy/du . du/dx
dy/dx = 2u . Sec²(x)
dy/dx = 2.Tan(x).Sec²(x)
dy/dx = 2.Tan(x) / Cos²(x)

y = Cot²(x) = [Cot(x)]²
y = u² > dy/du = 2u
u = Cot(x) > du/dv = –Csc²(x)
dy/dx = dy/du . du/dx
dy/dx = 2u . –Csc²(x)
dy/dx = 2.Cot(x) . –Csc²(x)
dy/dx = –2 / Tan(x).Sin²(x)

y = Tan²(ax) = [Tan(ax)]²
y = u² > dy/du = 2u
u = Tan(v) > du/dv = Sec²(v)
v = ax > dv/dx = a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec²(v) . a
dy/dx = 2.Tan(v) . Sec²(v) . a
dy/dx = 2.Tan(ax) . Sec²(ax) . a
= 2a . Tan(ax) . Sec²(ax)
dy/dx = 2a.Tan(ax) / Cos²(ax)

y = Cot²(ax) = [Cot(ax)]²
y = u² > dy/du = 2u
u = Cot(v) > du/dv = –Csc²(v)
v = ax > dv/dx = a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc²(v) . a
dy/dx = 2.Cot(v) . –Csc²(v) . a
dy/dx = 2.Cot(ax) . –Csc²(ax) . a
= 2a . Cot(ax) . –Csc²(ax)
dy/dx = –2a / Tan(ax).Sin²(ax)

y = Tan²(x/a) = [Tan(x/a)]²
y = u² > dy/du = 2u
u = Tan(v) > du/dv = Sec²(v)
v = x/a > dv/dx = 1/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec²(v) / a
dy/dx = 2.Tan(v) . Sec²(v) / a
dy/dx = 2.Tan(x/a) . Sec²(x/a) / a
dy/dx = 2.Tan(x/a) / a.Cos²(x/a)

y = Cot²(x/a) = [Cot(x/a)]²
y = u² > dy/du = 2u
u = Cot(v) > du/dv = –Csc²(v)
v = x/a > dv/dx = 1/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc²(v) / a
dy/dx = 2.Cot(v) . –Csc²(v) / a
dy/dx = 2.Cot(x/a) . –Csc²(x/a) / a
dy/dx = –2 / a.Tan(x/a).Sin²(x/a)

y = Tan²(a/x) = [Tan(a/x)]²
y = u² > dy/du = 2u
u = Tan(v) > du/dv = Sec²(v)
v = a/x > dv/dx = –a/x²
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec²(v) . –a/x²
dy/dx = 2.Tan(v) . Sec²(v) . –a/x²
dy/dx = 2.Tan(a/x) . Sec²(a/x) . –a/x²
= –2a . Tan(a/x) . Sec²(a/x) / x²
dy/dx = –2a.Tan(a/x) / x².Cos²(a/x)

y = Cot²(a/x) = [Cot(a/x)]²
y = u² > dy/du = 2u
u = Cot(v) > du/dv = –Csc(v)
v = a/x > dv/dx = –a/x²
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc²(v) . –a/x²
dy/dx = 2.Cot(v) . –Csc²(v) / a
dy/dx = 2.Cot(a/x) . –Csc²(a/x) . –a/x²
= 2a . Cot(a/x) . –Csc²(a/x) / x²
dy/dx = 2a / x².Tan(a/x).Sin²(a/x)

y = Tan²(x²) = [Tan(x²)]²
y = u² > dy/du = 2u
u = Tan(v) du/dv = Sec²(v)
v = x² > dv/dx = 2x
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec²(v) . 2x
dy/dx = 2.Tan(v) . Sec²(v) . 2x
dy/dx = 2.Tan(x²) . Sec²(x²) . 2x
= 4x . Tan(x²) . Sec²(x²)
dy/dx = 4x.Tan(x²) / Cos²(x²)

y = Cot²(x²) = [Cot(x²)]²
y = u² > dy/du = 2u
u = Cot(v) > du/dv = –Csc²(v)
v = x² > dv/dx = 2x
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc²(v) . 2x
dy/dx = 2.Cot(v) . –Csc²(v) . 2x
dy/dx = 2.Cot(x²) . –Csc²(x²) . 2x
= 4x . Cot(x²) . –Csc²(x²)
dy/dx = –4x / Tan(x²).Sin²(x²)

y = Tan²(ax²) = [Tan(ax²)]²
y = u² > dy/du = 2u
u = Tan(v) > du/dv = Sec²(v)
v = ax² > dv/dx = 2ax
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec²(v) . 2ax
dy/dx = 2.Tan(v) . Sec²(v) . 2ax
dy/dx = 2.Tan(ax²) . Sec²(ax²) . 2ax
= 4ax . Tan(ax²) . Sec²(ax²)
dy/dx = 4ax.Tan(ax²) / Cos²(ax²)

y = Cot²(ax²) = [Cot(ax²)]²
y = u² > dy/du = 2u
u = Cot(v) > du/dv = –Csc²(v)
v = ax² > dv/dx = 2ax
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc²(v) . 2ax
dy/dx = 2.Cot(v) . –Csc²(v) . 2x
dy/dx = 2.Cot(ax²) . –Csc²(ax²) . 2ax
= 4ax . Cot(ax²) . Csc(ax²)
dy/dx = –4ax / Tan(ax²).Sin²(ax²)

y = Tan²(x²/a) = [Tan(x²/a)]²
y = u² > dy/du = 2u
u = Tan(v) > du/dv = Sec²(v)
v = x²/a > dv/dx = 2x/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec²(v) . 2x/a
dy/dx = 2.Tan(v) . Sec²(v) . 2x/a
dy/dx = 2.Tan(x²/a) . Sec²(x²/a) . 2x/a
= 4x . Tan(x²/a) . Sec²(x²/a) /a
dy/dx = 4x.Tan(x²/a) / a.Cos²(x²/a)

y = u² > dy/du = 2u
u = Cot(v) > du/dv = –Csc²(v)
v = x²/a > dv/dx = 2x/a
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc²(v) . 2x/a
dy/dx = 2.Cot(v) . –Csc²(v) . 2x/a
dy/dx = 2.Cot(x²/a) . –Csc²(x²/a) . 2x/a
= 4x . Cot(x²/a) . –Csc²(x²/a) /a
dy/dx = –4x / a.Tan(x²/a).Sin²(x²/a)

y = Tan²(a/x²) = [Tan(a/x²)]²
y = u² > dy/du = 2u
u = Tan(v) > du/dv = Sec²(v)
v = a/x² > dv/dx = –2a/x³
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . Sec²(v) . –2a/x³
dy/dx = 2.Tan(v) . Sec²(v) . –2a/x³
dy/dx = 2.Tan(a/x²) . Sec²(a/x²) . –2a/x³
= –4a . Tan(a/x²) . Sec²(a/x²) / x³
dy/dx = –4a.Tan(a/x²) / x³.Cos²(a/x²)

y = Cot²(a/x²) = [Cot(a/x²)]²
y = u² > dy/du = 2u
u = Cot(v) > du/dv = –Csc²(v)
v = a/x² > dv/dx = –2a/x³
dy/dx = dy/du . du/dv . dv/dx
dy/dx = 2u . –Csc²(v) . –2a/x³
dy/dx = 2.Cot(v) . –Csc²(v) . –2a/x³
dy/dx = 2.Cot(a/x²) . –Csc²(a/x²) . –2a/x³
= 4a . Cot(a/x²) . Csc²(a/x²) / x³
dy/dx = 4a / x³.Tan(a/x²).Sin²(a/x²)

Further Reading

You will find further reading on this subject in reference publications(19)

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