Derivative Rules

Basic rules

Constant

\frac{d}{d x} [c] = 0

Power rule

\frac{d}{d x} [x^{n}] = n x^{n - 1}

Constant multiple

\frac{d}{d x} [c f (x)] = c f^{'} (x)

Sum and difference

\frac{d}{d x} [f (x) \pm g (x)] = f^{'} (x) \pm g^{'} (x)

Product rule

\frac{d}{d x} [f (x) g (x)] = f^{'} (x) g (x) + f (x) g^{'} (x)

Quotient rule

\frac{d}{d x} [\frac{f (x)}{g (x)}] = \frac{f^{'} (x) g (x) - f (x) g^{'} (x)}{[g (x)]^{2}}

Chain rule

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

Inverse function

(f^{- 1})^{'} (b) = \frac{1}{f^{'} (f^{- 1} (b))}

e to the x

\frac{d}{d x} [e^{x}] = e^{x}

a to the x

\frac{d}{d x} [a^{x}] = a^{x} \ln a

Natural log

\frac{d}{d x} [\ln | x |] = \frac{1}{x}

Log base a

\frac{d}{d x} [\log_{a} x] = \frac{1}{x \ln a}

Sine

\frac{d}{d x} [\sin x] = \cos x

Cosine

\frac{d}{d x} [\cos x] = - \sin x

Tangent

\frac{d}{d x} [\tan x] = \sec^{2} x

Cotangent

\frac{d}{d x} [\cot x] = - \csc^{2} x

Secant

\frac{d}{d x} [\sec x] = \sec x \tan x

Cosecant

\frac{d}{d x} [\csc x] = - \csc x \cot x

Inverse sine

\frac{d}{d x} [\arcsin x] = \frac{1}{\sqrt{1 - x^{2}}}

Inverse cosine

\frac{d}{d x} [\arccos x] = - \frac{1}{\sqrt{1 - x^{2}}}

Inverse tangent

\frac{d}{d x} [\arctan x] = \frac{1}{1 + x^{2}}

Inverse secant

\frac{d}{d x} [arcsec x] = \frac{1}{| x | \sqrt{x^{2} - 1}}

Trouble with a formula often traces back to an earlier skill. Our free check-in finds the exact gap. A tutor then works on it with your child.