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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)

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