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

Graphical or geometric meaning

Every tangent slope of $e^x$ equals the point's height. For $e^{-3x}$, values remain positive but decrease, and the factor $-3$ makes the initial decay three times as steep as that of $e^{-x}$.

Common mistakes and why they fail

Verification and reasonableness checks

• For $a>1$, verify that $a^x\ln a$ is positive.
• Use logarithmic identities or a numerical difference quotient at one point.
• Factor the derivative and check that every product-rule term is present.

Separate the base from the exponent

For $a>0$, $\frac d{dx}a^{u(x)}=a^{u(x)}\ln(a)u'(x)$, whereas $\frac d{dx}e^{u(x)}=e^{u(x)}u'(x)$. The factor $\ln a$ disappears only for base $e$. If both base and exponent vary, as in $u(x)^{v(x)}$, logarithmic differentiation is safer and requires attention to the real-valued domain. A qualitative check uses sign: a positive-base exponential never changes sign, and its derivative has the sign of $\ln(a)u'$. Thus $0<a<1$ reverses the growth direction associated with $u'$. Testing a simple linear exponent can confirm that every chain-rule factor was retained.

Practice

1. Differentiate $e^{5x}$.
2. Differentiate $2^x$.
3. Differentiate $e^{x^2+1}$.

Connections and next steps

• Chain Rule
• Derivatives of Logarithmic Functions
• Related Rates