Table of Derivatives

$$\frac{d}{dx}[af(x) + bg(x)] = a f'(x) + b g'(x)$$

where $a$ and $b$ are constants.

$$\frac{d}{dx}(c) = 0$$

For $n \in \mathbb{R}$:

$$\frac{d}{dx}(x^n) = n x^{n-1}$$

Generalized version (chain rule):

$$\frac{d}{dx}(u^n) = n u^{n-1} \frac{du}{dx}$$

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

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}, \quad g(x) \neq 0$$

Special case:

$$\frac{d}{dx}\left[\frac{1}{f(x)}\right] = -\frac{f'(x)}{[f(x)]^2}, \quad f(x) \neq 0$$

$$\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$$

If $y = f(x)$ has a differentiable inverse $f^{-1}$:

$$\frac{d}{dx}f^{-1}(x) = \frac{1}{f'(f^{-1}(x))}$$

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$$\begin{aligned} \frac{d}{dx}(x) &= 1 \\ \frac{d}{dx}(\sqrt{x}) &= \frac{1}{2\sqrt{x}}, \quad x > 0 \\ \frac{d}{dx}(\sqrt[n]{x}) &= \frac{1}{n\sqrt[n]{x^{n-1}}}, \quad x \neq 0 \\ \frac{d}{dx}\left(\frac{1}{x}\right) &= -\frac{1}{x^2}, \quad x \neq 0 \\ \frac{d}{dx}\left(\frac{1}{x^n}\right) &= -\frac{n}{x^{n+1}}, \quad x \neq 0 \end{aligned}$$

$$\begin{aligned} \frac{d}{dx}(|x|) &= \operatorname{sgn}(x), \quad x \neq 0 \\ \frac{d}{dx}(\operatorname{sgn}(x)) &= 0, \quad x \neq 0 \end{aligned}$$

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$$\frac{d}{dx}(e^x) = e^x$$

For $a > 0$, $a \neq 1$:

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

$$\frac{d}{dx}(\ln x) = \frac{1}{x}, \quad x > 0$$

For $a > 0$, $a \neq 1$:

$$\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}, \quad x > 0$$

For $f(x) > 0$:

$$\frac{d}{dx}[f(x)^{g(x)}] = f(x)^{g(x)} \left[ g'(x) \ln f(x) + g(x) \frac{f'(x)}{f(x)} \right]$$

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$$\begin{aligned} \frac{d}{dx}(\sin x) &= \cos x \\ \frac{d}{dx}(\cos x) &= -\sin x \\ \frac{d}{dx}(\tan x) &= \sec^2 x = 1 + \tan^2 x \\ \frac{d}{dx}(\cot x) &= -\csc^2 x = -(1 + \cot^2 x) \\ \frac{d}{dx}(\sec x) &= \sec x \tan x \\ \frac{d}{dx}(\csc x) &= -\csc x \cot x \end{aligned}$$

$$\begin{aligned} \frac{d}{dx}(\arcsin x) &= \frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1 \\ \frac{d}{dx}(\arccos x) &= -\frac{1}{\sqrt{1 - x^2}}, \quad |x| < 1 \\ \frac{d}{dx}(\arctan x) &= \frac{1}{1 + x^2} \\ \frac{d}{dx}(\operatorname{arccot} x) &= -\frac{1}{1 + x^2} \\ \frac{d}{dx}(\operatorname{arcsec} x) &= \frac{1}{|x|\sqrt{x^2 - 1}}, \quad |x| > 1 \\ \frac{d}{dx}(\operatorname{arccsc} x) &= -\frac{1}{|x|\sqrt{x^2 - 1}}, \quad |x| > 1 \end{aligned}$$

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$$\begin{aligned} \frac{d}{dx}(\sinh x) &= \cosh x \\ \frac{d}{dx}(\cosh x) &= \sinh x \\ \frac{d}{dx}(\tanh x) &= \operatorname{sech}^2 x = 1 - \tanh^2 x \\ \frac{d}{dx}(\coth x) &= -\operatorname{csch}^2 x = 1 - \coth^2 x \\ \frac{d}{dx}(\operatorname{sech} x) &= -\operatorname{sech} x \tanh x \\ \frac{d}{dx}(\operatorname{csch} x) &= -\operatorname{csch} x \coth x \end{aligned}$$

$$\begin{aligned} \frac{d}{dx}(\operatorname{arsinh} x) &= \frac{1}{\sqrt{x^2 + 1}} \\ \frac{d}{dx}(\operatorname{arcosh} x) &= \frac{1}{\sqrt{x^2 - 1}}, \quad x > 1 \\ \frac{d}{dx}(\operatorname{artanh} x) &= \frac{1}{1 - x^2}, \quad |x| < 1 \\ \frac{d}{dx}(\operatorname{arcoth} x) &= \frac{1}{1 - x^2}, \quad |x| > 1 \\ \frac{d}{dx}(\operatorname{arsech} x) &= -\frac{1}{x\sqrt{1 - x^2}}, \quad 0 < x < 1 \\ \frac{d}{dx}(\operatorname{arcsch} x) &= -\frac{1}{|x|\sqrt{1 + x^2}}, \quad x \neq 0 \end{aligned}$$

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$$\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t)\,dt = f(x,b(x)) \cdot b'(x) - f(x,a(x)) \cdot a'(x) + \int_{a(x)}^{b(x)} \frac{\partial f}{\partial x}(x,t)\,dt$$

1. Constant limits:

   $$\frac{d}{dx} \int_a^b f(x,t)\,dt = \int_a^b \frac{\partial f}{\partial x}(x,t)\,dt$$

2. Variable upper limit:

   $$\frac{d}{dx} \int_a^{g(x)} f(t)\,dt = f(g(x)) \cdot g'(x)$$

3. Variable lower limit:

   $$\frac{d}{dx} \int_{h(x)}^b f(t)\,dt = -f(h(x)) \cdot h'(x)$$

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$$\frac{d}{dx} \Gamma(x) = \Gamma(x) \psi(x)$$

where $\psi(x)$ is the digamma function.

$$\begin{aligned} \frac{d}{dx}[x^n J_n(x)] &= x^n J_{n-1}(x) \\ \frac{d}{dx}[x^{-n} J_n(x)] &= -x^{-n} J_{n+1}(x) \end{aligned}$$