# Differentiation Rules Reference Page

The derivative of $$f(x)$$ with respect to $$x$$ is denoted by $$f'(x)$$ where
$$f'(x) = \lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}.$$
Equivalent notation for $$f'(x)$$ include:
$$y’, \frac{df}{dx}, \frac{dy}{dx}, \frac{d}{dx}[f(x)]$$
Basic Differentiation Rules: Let $$c$$ be a constant.

• $$\displaystyle \frac{d}{dx}[c] = 0$$
• $$\displaystyle \frac{d}{dx}[x^n] = nx^{n-1}$$
• $$\displaystyle \frac{d}{dx}[e^x] = e^x$$
• $$\displaystyle \frac{d}{dx}[a^x] = a^x \ln a$$
• $$\displaystyle \frac{d}{dx}[\sin x] = \cos x$$
• $$\displaystyle \frac{d}{dx}[\csc x] = -\csc x \cot x$$
• $$\displaystyle \frac{d}{dx}[\cos x] = -\sin x$$
• $$\displaystyle \frac{d}{dx}[\sec x] = \sec x \tan x$$
• $$\displaystyle \frac{d}{dx}[\tan x] = \sec^2 x$$
• $$\displaystyle \frac{d}{dx}[\cot x] = -\csc^2 x$$
Derivative of a General Inverse Function:
$$\frac{d}{dx}[f^{-1}(x)] = \frac{1}{f'(f^{-1}(x))}$$
More Basic Differentiation Rules:

• $$\displaystyle \frac{d}{dx}[\ln x] = \frac{1}{x}$$
• $$\displaystyle \frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$$
• $$\displaystyle \frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1-x^2}}$$
• $$\displaystyle \frac{d}{dx}[\sec^{-1} x] = \frac{1}{|x|\sqrt{x^2-1}}$$
• $$\displaystyle \frac{d}{dx}[\cos^{-1} x] = \frac{-1}{\sqrt{1-x^2}}$$
• $$\displaystyle \frac{d}{dx}[\csc^{-1} x] = \frac{-1}{|x|\sqrt{x^2-1}}$$
• $$\displaystyle \frac{d}{dx}[\tan^{-1} x] = \frac{1}{1+x^2}$$
• $$\displaystyle \frac{d}{dx}[\cot^{-1} x] = \frac{-1}{1+x^2}$$
Properties of Derivatives:If $$f(x)$$ and $$g(x)$$ are differentiable at $$x$$ and $$c$$ is a constant, then

• Sum/Difference Rule:
$$\frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)]$$
• Product Rule:
$$\frac{d}{dx}[f(x) \cdot g(x)] = \frac{d}{dx}[f(x)] \cdot g(x) + \frac{d}{dx}[g(x)] \cdot f(x)$$
$$\frac{d}{dx}[f(x) \cdot g(x)] = \frac{d}{dx}[f(x)] \cdot g(x) + \frac{d}{dx}[g(x)] \cdot f(x)$$
\begin{aligned} &\frac{d}{dx}[f(x) \cdot g(x)] \\ & \qquad = \frac{d}{dx}[f(x)] \cdot g(x) + \frac{d}{dx}[g(x)] \cdot f(x) \end{aligned}
• Quotient Rule:
$$\frac{d}{dx}\left[ \frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot\frac{d}{dx}[f(x)] – \frac{d}{dx}[g(x)] \cdot f(x)}{(g(x))^2}$$
$$\frac{d}{dx}\left[ \frac{f(x)}{g(x)}\right] = \frac{g(x) \cdot\frac{d}{dx}[f(x)] – \frac{d}{dx}[g(x)] \cdot f(x)}{(g(x))^2}$$
\begin{aligned} & \frac{d}{dx}\left[ \frac{f(x)}{g(x)}\right] \\ & \qquad = \frac{g(x) \cdot\frac{d}{dx}[f(x)] – \frac{d}{dx}[g(x)] \cdot f(x)}{(g(x))^2} \end{aligned}
• Constant Multiplier Rule:
$$\frac{d}{dx}[c \cdot f(x)] = c \cdot \frac{d}{dx}[f(x)]$$
• Chain Rule: If, in addition, $$f(x)$$ is differentiable at $$g(x)$$ then
$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot \frac{d}{dx}[g(x)]$$