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Calculus Differentiation

Derivatives of exponential and log functions

20 practice questions 0 video lessons Theory + worked examples

Derivatives of Exponential and Log Functions

California Calculus • Standard 4.4 • Differentiation

Derivatives of Exponential and Log Functions is a topic in Differentiation in the California Calculus Standards. It is aligned to Standard 4.4, which requires students to derive and use the derivative formulas for exponential and logarithmic functions.

The exponential and logarithmic derivatives are \(\dfrac{d}{dx}e^x=e^x\) and \(\dfrac{d}{dx}\ln x=\dfrac{1}{x}\), extended to \(e^{g(x)}\), \(\ln(g(x))\), and the general base \(a^x\) with the chain rule.

California Calculus › Differentiation › Derivatives of Exponential and Log Functions  —  Standard 4.4

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Theory

The exponential and logarithm derivatives are remarkably clean: \(\dfrac{d}{dx}e^{x}=e^{x}\) (its own derivative) and \(\dfrac{d}{dx}\ln x=\dfrac{1}{x}\). With the chain rule they extend to \(e^{kx}\), \(\ln(g(x))\), and any base \(a^{x}\).

Two special derivatives anchor this topic:

\[\dfrac{d}{dx}e^{x}=e^{x},\qquad\dfrac{d}{dx}\ln x=\dfrac{1}{x}\]

The exponential \(e^{x}\) is the one function equal to its own derivative — its slope at any point equals its height.

With the chain rule, \(\dfrac{d}{dx}e^{g(x)}=e^{g(x)}g'(x)\) and \(\dfrac{d}{dx}\ln(g(x))=\dfrac{g'(x)}{g(x)}\).

Key idea: for a base other than \(e\), a natural-log factor appears: \(\dfrac{d}{dx}a^{x}=a^{x}\ln a\).
The exponential curve whose slope equals its height For e to the x the tangent slope at any point equals the height of the curve there, because the derivative of e to the x is itself. x y slope = e^x
\(\dfrac{d}{dx}e^{x}=e^{x}\): the slope equals the height.
The natural log curve with a tangent of slope one over x The derivative of ln x is one over x, so the tangent gets shallower as x grows. x y
\(\dfrac{d}{dx}\ln x=\dfrac{1}{x}\): the tangent flattens as \(x\) grows.

The base facts and their chain-rule forms:

\[\dfrac{d}{dx}e^{g(x)}=e^{g(x)}\,g^{\prime}(x),\qquad\dfrac{d}{dx}\ln\big(g(x)\big)=\dfrac{g^{\prime}(x)}{g(x)}\]

derivative of e to the g is e to the g times g prime; derivative of ln g is g prime over g

The general-base exponential:

\[\dfrac{d}{dx}a^{x}=a^{x}\ln a\]

derivative of a to the x is a to the x times ln a

Note: because \(\ln e=1\), the general rule reduces to \(\dfrac{d}{dx}e^{x}=e^{x}\).

How to differentiate exp/log expressions

  1. Spot the outer function: \(e^{(\cdots)}\), \(\ln(\cdots)\), or \(a^{x}\).
  2. Apply the base rule, keeping the inside unchanged for exponentials.
  3. Multiply by the inner derivative (for \(\ln\), divide by the inside); use the product rule when \(x\) multiplies \(e^x\) or \(\ln x\).
Example 1 — Exponential chain rule
Differentiate \(y=e^{2x}\).
Solution

Outer \(e^{u}\), inner \(2x\).

\(y'\)\(=\)\(e^{2x}\cdot 2=2e^{2x}\)

derivative is 2 e to the 2x

Example 2 — Log chain rule
Differentiate \(y=\ln(x^2+1)\).
Solution

Use \(\dfrac{d}{dx}\ln(g)=\dfrac{g'}{g}\).

\(y'\)\(=\)\(\dfrac{2x}{x^2+1}\)

derivative is 2x over x squared plus one

Example 3 — Product with \(e^x\)
Differentiate \(y=x\,e^{x}\).
Solution

Product rule with \(u=x\), \(v=e^{x}\).

\(y'\)\(=\)\(1\cdot e^{x}+x\,e^{x}\)
\(=\)\((x+1)e^{x}\)

derivative is x plus one times e to the x

Example 4 — A general base
Differentiate \(y=3^{x}\).
Solution

Use \(\dfrac{d}{dx}a^{x}=a^{x}\ln a\).

\(y'\)\(=\)\(3^{x}\ln 3\)

derivative is 3 to the x times natural log of 3

Common pitfalls

\(\ln\) divides, exp multiplies. \(\dfrac{d}{dx}\ln(g)=\dfrac{g'}{g}\), while \(\dfrac{d}{dx}e^{g}=e^{g}g'\).
Do not drop the \(\ln a\) for other bases. \(\dfrac{d}{dx}2^{x}=2^{x}\ln 2\), not \(2^{x}\).
\(e^{x}\) does not follow the power rule. The exponent is the variable, so \(\dfrac{d}{dx}e^{x}=e^{x}\), not \(x e^{x-1}\).

Frequently asked questions

What is the derivative of e^x?

\(\dfrac{d}{dx}e^{x}=e^{x}\). The exponential function is its own derivative; its slope equals its height everywhere.

What is the derivative of ln x?

\(\dfrac{d}{dx}\ln x=\dfrac{1}{x}\), for \(x>0\).

How do you differentiate ln of a function?

Use \(\dfrac{d}{dx}\ln(g(x))=\dfrac{g'(x)}{g(x)}\): the derivative of the inside over the inside.

What is the derivative of a^x for a base other than e?

\(\dfrac{d}{dx}a^{x}=a^{x}\ln a\). A factor of the natural log of the base appears.

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