Derivative Rules
Linearity (or the sum rule)
The derivative is linear, that is
$$\frac{d}{dx} \Big(C f(x) + g(x) \Big) = Cf'(x) + g'(x)$$
The Product and Quotient Rules
$$\frac{d}{dx} \Big(f(x)g(x) \Big) = f'(x)g(x) + f(x)g'(x)$$
$$\frac{d}{dx} \Big(\frac{f(x)}{g(x)} \Big) = \frac{f'(x)g(x) – f(x)g'(x)}{g^2(x)}$$
The Chain Rule
$$\frac{d}{dx} \Big( f(g(x))\Big) = f'(g(x))g'(x)$$
Derivatives of common functions
Derivatives of power functions and constants
- $\displaystyle \frac{d}{dx} \Big( C \Big) = 0$
- $\displaystyle \frac{d}{dx} \Big( x \Big) = 1$
- $\displaystyle \frac{d}{dx} \Big( x^p \Big) = p x^{p-1}$ for any real $p$
Special cases of power functions
- $\displaystyle \frac{d}{dx} \Big( \frac{1}{x^p} \Big) = -\frac{p}{x^{p+1}}$ using $\frac{1}{x^p} = x^{-p}$
- $\displaystyle\frac{d}{dx} \Big( \sqrt[p]{x} \Big) = \frac{1}{p} x^{\tfrac{1}{p}-1}$ using $\sqrt[p]{x} = x^{1/p}$
Derivatives of trig functions
- $\displaystyle \frac{d}{dx} \Big( \sin x \Big) = \cos x$
- $\displaystyle \frac{d}{dx} \Big( \cos x \Big) = -\sin x$
- $\displaystyle \frac{d}{dx} \Big( \tan x \Big) = \sec^2 x = \frac{1}{ \cos^2 x}$
- $\displaystyle\frac{d}{dx} \Big( \sec x \Big) = \sec x\ \tan x$
- $\displaystyle\frac{d}{dx} \Big( \csc x \Big) = -\csc x \ \cot x$
- $\displaystyle\frac{d}{dx} \Big( \cot x \Big) = -\csc^2 x = – \frac{1}{\sin^2 x}$
Derivatives of inverse trig functions
Notice that in this section $\sin^{-1} x$ denotes the inverse of $\sin x$ (and similarly for the other trig function). Inverses of trig functions are also sometimes denoted $\arcsin x, \arccos x$, etc.
- $\displaystyle \frac{d}{dx} \Big( \sin^{-1} x \Big) = \frac{1}{\sqrt{1-x^2}}$
- $\displaystyle \frac{d}{dx} \Big( \cos^{-1} x \Big) = -\frac{1}{\sqrt{1-x^2}}$
- $\displaystyle \frac{d}{dx} \Big( \tan^{-1} x \Big) = \frac{1}{1+x^2}$
- $\displaystyle\frac{d}{dx} \Big( \sec^{-1} x \Big) = \frac{1}{|x|\sqrt{x^2-1}}$
- $\displaystyle\frac{d}{dx} \Big( \csc^{-1} x \Big) = -\frac{1}{|x|\sqrt{x^2-1}}$
- $\displaystyle\frac{d}{dx} \Big( \cot^{-1} x \Big) =-\frac{1}{1+x^2}$
Derivatives of exponential and logarithmic functions
- $\displaystyle \frac{d}{dx} \Big( e^x \Big) = e^x$
- $\displaystyle \frac{d}{dx} \Big( \ln x \Big) = \frac{1}{x}$
- $\displaystyle\frac{d}{dx} \Big( a^x \Big) = a^x \ln a$
- $\displaystyle\frac{d}{dx} \Big( \log_a x \Big) = \frac{1}{x \ln a}$
Limit definition of the Derivative
The derivative at a point
$$f'(a) = \lim_{h\to 0} \frac{f(a+h)-f(a)}{h}$$
$$f'(a) = \lim_{x\to a} \frac{f(x)-f(a)}{x-a}$$
The derivative function
$$f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$
$$f'(x) = \lim_{s\to x} \frac{f(s)-f(x)}{s-x}$$