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) |
y = Sin(a/x) |
y = Cos(a/x) |
y = Sin(x²) |
y = Cos(x²) |
y = Sin(ax²) |
y = Cos(ax²) |
y = Sin(x²/a) |
y = Cos(x²/a) |
y = Sin(a/x²) |
y = Cos(a/x²) |
y = Sin²(x) = [Sin(x)]² |
y = Cos²(x) = [Cos(x)]² |
y = Sin²(ax) = [Sin(ax)]² |
y = Cos²(ax) = [Cos(ax)]² |
y = Sin²(x/a) = [Sin(x/a)]² |
y = Cos²(x/a) = [Cos(x/a)]² |
y = Sin²(a/x) = [Sin(a/x)]² |
y = Cos²(a/x) = [Cos(a/x)]² |
y = Sin²(x²) = [Sin(x²)]² |
y = Cos²(x²) = [Cos(x²)]² |
y = Sin²(ax²) = [Sin(ax²)]² |
y = Cos²(ax²) = [Cos(ax²)]² |
y = Sin²(x²/a) = [Sin(x²/a)]² |
y = Cos²(x²/a) = [Cos(x²/a)]² |
y = Sin²(a/x²) = [Sin(a/x²)]² |
y = Cos²(a/x²) = [Cos(a/x²)]² |
y = Csc(a/x) |
y = Sec(a/x) |
y = Csc(x²) |
y = Sec(x²) |
y = Csc(ax²) |
y = Sec(ax²) |
y = Csc(x²/a) |
y = Sec(x²/a) |
y = Csc(a/x²) |
y = Sec(a/x²) |
y = Csc²(x) = [Csc(x)]² |
y = Sec²(x) = [Sec(x)]² |
y = Csc²(ax) = [Csc(ax)]² |
y = Sec²(ax) = [Sec(ax)]² |
y = Csc²(x/a) = [Csc(x/a)]² |
y = Sec²(x/a) = [Sec(x/a)]² |
y = Csc²(a/x) = [Csc(a/x)]² |
y = Sec²(a/x) = [Sec(a/x)]² |
y = Csc²(x²) = [Csc(x²)]² |
y = Sec²(x²) = [Sec(x²)]² |
y = Csc²(ax²) = [Csc(ax²)]² |
y = Sec²(ax²) = [Sec(ax²)]² |
y = Csc²(x²/a) = [Csc(x²/a)]² |
y = Sec²(x²/a) = [Sec(x²/a)]² |
y = Csc²(a/x²) = [Csc(a/x²)]² |
y = Sec²(a/x²) = [Sec(a/x²)]² |
y = Tan(a/x) |
y = Cot(a/x) |
y = Tan(x²) |
y = Cot(x²) |
y = Tan(ax²) |
y = Cot(ax²) |
y = Tan(x²/a) |
y = Cot(x²/a) |
y = Tan(a/x²) |
y = Cot(a/x²) |
y = Tan²(x) = [Tan(x)]² |
y = Cot²(x) = [Cot(x)]² |
y = Tan²(ax) = [Tan(ax)]² |
y = Cot²(ax) = [Cot(ax)]² |
y = Tan²(x/a) = [Tan(x/a)]² |
y = Cot²(x/a) = [Cot(x/a)]² |
y = Tan²(a/x) = [Tan(a/x)]² |
y = Cot²(a/x) = [Cot(a/x)]² |
y = Tan²(x²) = [Tan(x²)]² |
y = Cot²(x²) = [Cot(x²)]² |
y = Tan²(ax²) = [Tan(ax²)]² |
y = Cot²(ax²) = [Cot(ax²)]² |
y = Tan²(x²/a) = [Tan(x²/a)]² |
y = u² > dy/du = 2u |
y = Tan²(a/x²) = [Tan(a/x²)]² |
y = Cot²(a/x²) = [Cot(a/x²)]² |
You will find further reading on this subject in reference publications(19)