Calculus Reference
Exponential and Logarithmic Derivatives
When the variable is in an exponent or logarithm, keep the function’s form and multiply by the derivative of its inside expression.
Solved Examples
Examples cover natural and common bases, composite exponents, and logarithmic inputs.
Derivative Formula Table
| Function | Derivative | Condition |
|---|---|---|
| $e^{u(x)}$ | $e^{u(x)}u'(x)$ | All real $u$. |
| $a^{u(x)}$ | $a^{u(x)}\ln(a)u'(x)$ | $a>0$. |
| $\ln|u(x)|$ | $\frac{u'(x)}{u(x)}$ | $u(x)\ne0$. |
| $\log_a|u(x)|$ | $\frac{u'(x)}{u(x)\ln(a)}$ | $a>0$, $a\ne1$, and $u(x)\ne0$. |
The absolute value in the logarithm formulas covers both positive and negative nonzero inputs. For $\ln(u)$ without absolute value, the original function requires $u>0$.
Exponential Functions
The natural exponential $e^u$ keeps its form when differentiated. A different positive base adds a factor of $\ln(a)$. If the exponent is a function, multiply by its derivative.
Example: Natural Exponential with a Composite Exponent
Differentiate $f(x)=e^{2x^2-1}$
The outside function is $e^u$ and the inside is $u=2x^2-1$.
- Keep the exponential and multiply by the inside derivative.
- Differentiate $2x^2-1$ and simplify.
Example: Exponential with a Common Base
Differentiate $g(x)=3^{x^2+1}$
Use the constant-base rule and remember its $\ln(3)$ factor.
- Keep $3^{x^2+1}$ and multiply by $\ln(3)$.
- Multiply by the derivative of the exponent, $2x$.
Logarithmic Functions
The natural logarithm has derivative $1/u$ with respect to its input. A composite input adds the chain-rule factor $u'$. A logarithm with base $a$ has an additional divisor of $\ln(a)$.
Example: Natural Logarithm with an Inside Function
Differentiate $h(x)=\ln(3x+2)$
The inside expression is positive when $x>-\frac23$.
- Divide the inside derivative by the inside expression.
- Simplify and keep the domain of the original logarithm.
Example: Logarithm with a Common Base
Differentiate $p(x)=\log_2(x^2+1)$
The quadratic input is positive for every real $x$.
- Use $u'/[u\ln(2)]$ for a base-$2$ logarithm.
- Substitute $u=x^2+1$ and $u'=2x$.