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# Common derivatives

## Polynomials

$$\frac{d}{dx}(c) = 0 \notag$$
$$\frac{d}{dx}(x)= 1 \notag$$
$$\frac{d}{dx}(ax) = a \notag$$
$$\frac{d}{dx}(x^n) = nx^{n-1} \notag$$
$$\frac{d}{dx}(cx^n) = ncx^{n-1} \notag$$
$$\frac{d}{dx}(ax^2 + bx + c) = 2ax + b \notag$$

## Trigonometric functions

$$\frac{d}{dx} \sin(x) = \cos(x) \notag$$
$$\frac{d}{dx} \cos(x) = -\sin(x) \notag$$
$$\frac{d}{dx} \tan(x) = \sec^2(x) \notag$$
$$\frac{d}{dx} \sec(x) = \sec (x) \times tan(x) \notag$$
$$\frac{d}{dx} \csc(x) = -csc(x) \times cot(x) \notag$$
$$\frac{d}{dx} \cot(x) = -csc^2(x) \notag$$

## Inverse trigonometric functions

$$\frac{d}{dx} \sin^{-1}(x) = \frac {1} { \sqrt { 1-x^2 } } \notag$$
$$\frac{d}{dx} \cos^{-1}(x) = - \frac {1} { \sqrt { 1-x^2 } } \notag$$
$$\frac{d}{dx} \tan^{-1}(x) = \frac {1} { 1 + x^2 } \notag$$
$$\frac{d}{dx} \sec^{-1}(x) = \frac {1} { \left| x \right| \sqrt { x^2 - 1 } } \notag$$
$$\frac{d}{dx} \csc^{-1}(x) = - \frac {1} { \left| x \right| \sqrt { x^2 -1 } } \notag$$
$$\frac{d}{dx} \cot^{-1}(x) = - \frac {1} { 1+x^2 } \notag$$

## Exponential functions

$$\frac{d}{dx} (a^x) = a \ln(a) \notag$$
$$\frac{d}{dx} (\mathrm{e}^x) = \mathrm{e}^x \notag$$

## Logarithm functions

$$\frac{d}{dx} \ln(x) = \frac{1}{x} \quad \forall \quad x>0 \notag$$
$$\frac{d}{dx} \ln( \left| x \right| ) = \frac{1}{x} \quad \forall \quad x \neq 0 \notag$$
$$\frac{d}{dx} \log_n(x) = \frac{1}{x \ln n} \quad \forall \quad x>0 \notag$$

## Hyperbolic trigonometric functions

$$\frac{d}{dx} \sinh(x) = \cosh(x) \notag$$
$$\frac{d}{dx} \cosh(x) = \sinh(x) \notag$$
$$\frac{d}{dx} \tanh(x) = \mathrm{sech}^2(x) \notag$$
$$\frac{d}{dx} \mathrm{ sech} (x) = - \mathrm{ sech(x) } \times \tanh(x) \notag$$
$$\frac{d}{dx} \mathrm{ csch }(x) = - \mathrm{ csch(x) } \times coth(x) \notag$$
$$\frac{d}{dx} \coth(x) = - \mathrm{ csch^2(x) } \notag$$