Differentiation Of Trigonometric Functions — Formula Sheet

\frac{d}{dx}(\sin x) = \cos x

\frac{d}{dx}(\cos x) = -\sin x

\frac{d}{dx}(\tan x) = \sec^2 x

\frac{d}{dx}(\cot x) = -\csc^2 x

\frac{d}{dx}(\sec x) = \sec x \tan x

\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1 - x^2}}

\frac{d}{dx}(\cos^{-1} x) = \frac{-1}{\sqrt{1 - x^2}}

\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1 + x^2}

\frac{d}{dx}(\cot^{-1} x) = \frac{-1}{1 + x^2}

\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2 - 1}}

\frac{d}{dx}(\sin u) = \cos u \cdot \frac{du}{dx}

\frac{d}{dx}(\tan u) = \sec^2 u \cdot \frac{du}{dx}

\frac{d}{dx}(\sin^{-1} u) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx}

\frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{dy}{dx}\right)

Using substitution and chain rule

d/dx(sin⁻¹x) = 1/√(1−x²), d/dx(cos⁻¹x) = −1/√(1−x²), d/dx(tan⁻¹x) = 1/(1+x²), and similar for others.

For example, if y = sin x, then dy/dx = cos x and d²y/dx² = −sin x.