Calculus Reference
Special Function Derivatives
Exponential, logarithmic, and trigonometric functions each have characteristic derivative rules.
Solved Examples
Jump directly to a worked example by function type.
Derivative Table
| Function | Derivative |
|---|---|
| $e^x$ | $e^x$ |
| $a^x$ | $a^x\ln a$ |
| $\ln x$ | $1/x$ |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2x$ |
Derivative of an Exponential Expression
Chain Rule with $e^x$
Find the derivative of the expression $f(x)=e^{3x^2}$.
Solution Steps
- Keep the exponential expression and differentiate its exponent.
- Multiply by the inside derivative and state the derivative: $f'(x)=6xe^{3x^2}$.
$$f'(x)=e^{3x^2}(6x)$$$$f'(x)=6xe^{3x^2}$$
Derivative of a Logarithmic Expression
Natural Logarithm
Find the derivative of the expression $g(x)=\ln(x^2+1)$.
Solution Steps
- Use $u'/u$ for the logarithm's inside expression and state the derivative: $g'(x)=\frac{2x}{x^2+1}$.
$$g'(x)=\frac{2x}{x^2+1}$$
Derivative of a Trigonometric Expression
Sine with an Inside Expression
Find the derivative of the expression $q(x)=\sin(2x)$.
Solution Steps
- Differentiate sine to cosine and keep the inside expression.
- Multiply by the inside derivative and state the derivative: $q'(x)=2\cos(2x)$.
$$q'(x)=\cos(2x)(2)$$$$q'(x)=2\cos(2x)$$