Derivatives of Exponential Functions

The central idea

The natural exponential satisfies $d(e^x)/dx=e^x$. For a constant base $a>0$, $a\ne1$, $d(a^x)/dx=a^x\ln a$. With a differentiable exponent $u(x)$, the chain rule gives $d(e^{u})/dx=e^{u}u'$ and $d(a^{u})/dx=a^{u}\ln(a)u'$.

Definitions, hypotheses, and notation

The factor $\ln a$ is forced by rewriting $a^x=e^{x\ln a}$. Differentiating this identity by the chain rule gives $a^x\ln a$, so the natural-base rule and general-base rule are not independent facts. When $0<a<1$, $\ln a<0$, explaining exponential decay without a separate derivative formula. Negative bases are excluded because $a^x$ is not real-valued and continuous for every real exponent.

In a model $P(t)=P_0e^{kt}$, differentiation yields $P'=kP$: growth rate is proportional to current amount. The sign of $k$ determines growth or decay, and $1/k$ sets a characteristic time scale. This relation provides an independent check on symbolic derivatives: after differentiating, the result should retain the exponential factor and have the expected sign.

Conceptual meaning

The function $e^x$ is uniquely normalized to equal its own instantaneous growth rate. Other bases grow in proportion to their current value, with constant of proportionality $\ln a$. Bases between zero and one have negative logarithm and therefore decay.

A dependable method and decision rule

1. Identify the entire exponent as the inner function $u$.
2. Copy the exponential factor unchanged.
3. Include $\ln a$ when the base is not $e$.
4. Multiply by $u'$ for a composite exponent.
5. Use product or quotient rules too if other factors surround the exponential.

Fully worked example

Common mistakes and why they fail