The formulas on this page give a recipe for computing derivatives. We'll justify some of these formulas in class, and the rest later in the semester.
Derivatives of Standard Functions
| Function $f(x)$ | Derivative $f'(x)$ | Comments |
| $c$ | $0$ | where $c$ is an arbitrary constant | $x^p$ | $px^{p-1}$ | where $p$ is an arbitrary constant | $\sin(x)$ | $\cos(x)$ | $\cos(x)$ | $-\sin(x)$ | don't forget the minus sign! | $\tan(x)$ | $\sec^2(x)$ | $b^x$ | $\ln(b) \, b^x$ | where $b$ is an arbitrary positive constant and $\ln(b)$ is the natural log of $b$. |
These functions can be combined with a couple of very simple rules:
Basic Differentiation Rules
| Function | Derivative | Comments |
| $f(x)+g(x)$ | $f'(x)+g'(x)$ | Derivative of a sum is a sum of derivatives | $cf(x)$ | $c f'(x)$ | Multiplicative constants come along for the ride |
Examples:
The derivative of $x^3$ (with respect to $x$) is $3x^2$, so by the
second rule the derivative of $8x^3$ is $8(3x^2)=24x^2$.
The derivative of
$-2x$ is $-2$ times the derivative of $x$, hence $-2(1)=-2$.
The derivative
of $3$ is 0.
Combining these with the first rule, we get that that derivative
of $8x^3-2x+3$ is $24x^2 - 2$.
The derivative of $\cos(x)$ is $-\sin(x)$, so the derivative of $4 \cos(x) + x$ is $-4 \sin(x) + 1$.
The derivative of $\tan(x)$ is $\sec^2(x)$ and the derivative of $5^x$ is $\ln(5) \, 5^x$, so the derivative of $\pi \tan(x) + 5^x$ is $\pi \, \sec^2(x) + \ln(5) \, 5^x$.