General Derivative Rules
Power Rule
$$\frac{d}{d x}\left(x^n\right)=n x^{n-1} $$
Sum Rule
$$\frac{d}{d x}[f(x)+g(x)]=f^{\prime}(x)+g^{\prime}(x)$$
Difference Rule
$$\frac{d}{d x}[f(x)-g(x)]=f^{\prime}(x)-g^{\prime}(x)$$
Product Rule
$$\frac{d}{d x}[f(x) g(x)]=f(x) \mathrm{g}^{\prime}(x)+g(x) f^{\prime}(x)$$
Quotient Rule
$$\frac{d}{d x}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x) f^{\prime}(x)-f(x)
g^{\prime}(x)}{[g(x)]^2}$$
Chain Rule
$$\frac{d}{d x} f(g(x))=f^{\prime}(g(x)) g^{\prime}(x)$$
Exponential & Logarithmic Derivative Rules
Exponential
$$\begin{aligned} & \frac{d}{d x}\left(e^x\right)=e^x \\ & \frac{d}{d
x}\left(a^x\right)=a^x \ln a \\ & \frac{d}{d x}\left(e^{g(x)}\right)=e^{g(x)}
g^{\prime}(x) \\ & \frac{d}{d x}\left(a^{g(x)}\right)=\ln (\mathrm{a}) \mathrm{a}^{g(x)}
g^{\prime}(x)\end{aligned}$$
Logarithmic
$$\begin{aligned} & \frac{d}{d x}(\ln x)=\frac{1}{x}, x>0 \\ & \frac{d}{d x} \ln
(g(x))=\frac{g^{\prime}(x)}{g(x)} \\ & \frac{d}{d x}\left(\log _a x\right)=\frac{1}{x
\ln a}, x>0 \\ & \frac{d}{d x}\left(\log _a g(x)\right)=\frac{g^{\prime}(x)}{g(x) \ln
a}\end{aligned}$$
Trig Derivative Rules
Trig
$$\begin{aligned} & \frac{d}{d x}(\sin x)=\cos x \\ & \frac{d}{d x}(\cos x)=-\sin x \\ &
\frac{d}{d x}(\tan x)=\sec ^2 x \\ & \frac{d}{d x}(\csc x)=-\csc x \cot x \\ &
\frac{d}{d x}(\sec x)=\sec x \tan x \\ & \frac{d}{d x}(\cot x)=-\csc ^2 x\end{aligned}$$
Inverse Trig
$$\begin{aligned} & \frac{d}{d x}\left(\sin ^{-1} x\right)=\frac{1}{\sqrt{1-x^2}}, x
\neq \pm 1 \\ & \frac{d}{d x}\left(\cos ^{-1} x\right)=\frac{-1}{\sqrt{1-x^2}}, x \neq
\pm 1 \\ & \frac{d}{d x}\left(\tan ^{-1} x\right)=\frac{1}{1+x^2} \\ & \frac{d}{d
x}\left(\cot ^{-1} x\right)=\frac{-1}{1+x^2} \\ & \frac{d}{d x}\left(\sec ^{-1}
x\right)=\frac{1}{x \sqrt{x^2-1}}, x \neq \pm 1,0 \\ & \frac{d}{d x}\left(\csc ^{-1}
x\right)=\frac{-1}{x \sqrt{x^2-1}}, x \neq \pm 1,0\end{aligned}$$