Calculus Tables
Derivative and integral tables.
This page contains tables of derivatives and integrals most commonly encountered in Calculus I - III. Other, more specialized tables may be found in other pages.
d d x c = 0 \begin{align}
\dv{}{x}c = 0
\end{align} d x d c = 0 d d x [ f ( x ) + g ( x ) ] = f ′ ( x ) + g ′ ( x ) \begin{align}
\dv{}{x}\left[f(x)+g(x)\right] = f'(x) + g'(x)
\end{align} d x d [ f ( x ) + g ( x ) ] = f ′ ( x ) + g ′ ( x ) d d x [ f ( x ) g ( x ) ] = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) \begin{align}
\dv{}{x}\left[f(x)g(x)\right] = f'(x)g(x) + f(x)g'(x)
\end{align} d x d [ f ( x ) g ( x ) ] = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) d d x f ( x ) g ( x ) = g ( x ) f ′ ( x ) − f ( x ) g ′ ( x ) [ g ( x ) ] 2 \begin{align}
\dv{}{x}\frac{f(x)}{g(x)} = \frac{g(x)f'(x)-f(x)g'(x)}{[g(x)]^2}
\end{align} d x d g ( x ) f ( x ) = [ g ( x ) ] 2 g ( x ) f ′ ( x ) − f ( x ) g ′ ( x ) d d x f ( g ( x ) ) = f ′ ( g ( x ) ) g ′ ( x ) \begin{align}
\dv{}{x}f(g(x)) = f'(g(x))g'(x)
\end{align} d x d f ( g ( x )) = f ′ ( g ( x )) g ′ ( x ) d d x x n = n x n − 1 \begin{align}
\dv{}{x}x^n = nx^{n-1}
\end{align} d x d x n = n x n − 1 Exponential and Logarithmic Functions
d d x e x = e x \begin{align}
\dv{}{x} e^x = e^x
\end{align} d x d e x = e x d d x a x = a x ln a , a > 0 \begin{align}
\dv{}{x} a^x = a^x \ln a,\quad a > 0
\end{align} d x d a x = a x ln a , a > 0 d d x ln ∣ x ∣ = 1 x , ∣ x ∣ ≠ 0 \begin{align}
\dv{}{x} \ln |x| = \frac{1}{x},\quad |x| \neq 0
\end{align} d x d ln ∣ x ∣ = x 1 , ∣ x ∣ = 0 d d x log a x = 1 x ln a , a > 0 \begin{align}
\dv{}{x} \log_a x = \frac{1}{x \ln a},\quad a > 0
\end{align} d x d log a x = x ln a 1 , a > 0 Trigonometric Functions
d d x sin x = cos x \begin{align}
\dv{}{x} \sin x = \cos x
\end{align} d x d sin x = cos x d d x cos x = − sin x \begin{align}
\dv{}{x} \cos x = -\sin x
\end{align} d x d cos x = − sin x d d x tan x = sec 2 x \begin{align}
\dv{}{x} \tan x = \sec^2 x
\end{align} d x d tan x = sec 2 x d d x csc x = − csc x cot x \begin{align}
\dv{}{x} \csc x = -\csc x \cot x
\end{align} d x d csc x = − csc x cot x d d x sec x = sec x tan x \begin{align}
\dv{}{x} \sec x = \sec x \tan x
\end{align} d x d sec x = sec x tan x d d x cot x = − csc 2 x \begin{align}
\dv{}{x} \cot x = -\csc^2 x
\end{align} d x d cot x = − csc 2 x Inverse Trigonometric Functions
d d x sin − 1 x = 1 1 − x 2 \begin{align}
\dv{}{x} \sin^{-1} x = \frac{1}{\sqrt{1-x^2}}
\end{align} d x d sin − 1 x = 1 − x 2 1 d d x cos − 1 x = − 1 1 − x 2 \begin{align}
\dv{}{x} \cos^{-1} x = -\frac{1}{\sqrt{1-x^2}}
\end{align} d x d cos − 1 x = − 1 − x 2 1 d d x tan − 1 x = 1 1 + x 2 \begin{align}
\dv{}{x} \tan^{-1} x = \frac{1}{1+x^2}
\end{align} d x d tan − 1 x = 1 + x 2 1 d d x csc − 1 x = − 1 ∣ x ∣ x 2 − 1 \begin{align}
\dv{}{x} \csc^{-1} x = -\frac{1}{|x|\sqrt{x^2-1}}
\end{align} d x d csc − 1 x = − ∣ x ∣ x 2 − 1 1 d d x sec − 1 x = 1 ∣ x ∣ x 2 − 1 \begin{align}
\dv{}{x} \sec^{-1} x = \frac{1}{|x|\sqrt{x^2-1}}
\end{align} d x d sec − 1 x = ∣ x ∣ x 2 − 1 1 d d x cot − 1 x = − 1 1 + x 2 \begin{align}
\dv{}{x} \cot^{-1} x = -\frac{1}{1+x^2}
\end{align} d x d cot − 1 x = − 1 + x 2 1 Hyperbolic Functions
d d x sinh x = cosh x \begin{align}
\dv{}{x} \sinh x = \cosh x
\end{align} d x d sinh x = cosh x d d x cosh x = sinh x \begin{align}
\dv{}{x} \cosh x = \sinh x
\end{align} d x d cosh x = sinh x d d x tanh x = s e c h 2 x \begin{align}
\dv{}{x} \tanh x = \sech^2 x
\end{align} d x d tanh x = sech 2 x d d x c s c h x = − c s c h x coth x \begin{align}
\dv{}{x} \csch x = -\csch x \coth x
\end{align} d x d csch x = − csch x coth x d d x s e c h x = − s e c h x tanh x \begin{align}
\dv{}{x} \sech x = -\sech x \tanh x
\end{align} d x d sech x = − sech x tanh x Inverse Hyperbolic Functions
d d x sinh − 1 x = 1 1 + x 2 \begin{align}
\dv{}{x} \sinh^{-1} x = \frac{1}{\sqrt{1+x^2}}
\end{align} d x d sinh − 1 x = 1 + x 2 1 d d x cosh − 1 x = 1 x 2 − 1 \begin{align}
\dv{}{x} \cosh^{-1} x = \frac{1}{\sqrt{x^2-1}}
\end{align} d x d cosh − 1 x = x 2 − 1 1 d d x tanh − 1 x = 1 1 − x 2 \begin{align}
\dv{}{x} \tanh^{-1} x = \frac{1}{1-x^2}
\end{align} d x d tanh − 1 x = 1 − x 2 1 d d x c s c h − 1 x = − 1 ∣ x ∣ 1 + x 2 \begin{align}
\dv{}{x} \csch^{-1} x = -\frac{1}{|x|\sqrt{1+x^2}}
\end{align} d x d csch − 1 x = − ∣ x ∣ 1 + x 2 1 d d x s e c h − 1 x = − 1 x 1 − x 2 \begin{align}
\dv{}{x} \sech^{-1} x = -\frac{1}{x\sqrt{1-x^2}}
\end{align} d x d sech − 1 x = − x 1 − x 2 1 d d x coth − 1 x = 1 1 − x 2 \begin{align}
\dv{}{x} \coth^{-1} x = \frac{1}{1-x^2}
\end{align} d x d coth − 1 x = 1 − x 2 1 Higher-Order Derivatives
General Leibniz rule: If f ( x ) f(x) f ( x ) g ( x ) g(x) g ( x ) n n n
d n d x n [ f ( x ) g ( x ) ] = ∑ k = 0 n ( n k ) d n − k d x n − k f ( x ) d k d x k g ( x ) . \begin{align}
\ndv{n}{}{x}[f(x)g(x)] = \sum_{k=0}^n \binom{n}{k} \ndv{n-k}{}{x}f(x)\ndv{k}{}{x}g(x).
\end{align} d x n d n [ f ( x ) g ( x )] = k = 0 ∑ n ( k n ) d x n − k d n − k f ( x ) d x k d k g ( x ) .