Derivatives of Exponential and Logarithmic Functions

Exponential and Logarithmic Derivatives

By the end of this lesson, you will be able to differentiate $e^x$, $a^x$, $\ln x$, and $\log_a x$, and apply logarithmic differentiation.

The Natural Exponential Function

$$\frac{d}{dx}[e^x] = e^x$$

This is the remarkable property of $e$: it is its own derivative. This is why $e$ is the "natural" base.

With the chain rule: $$\frac{d}{dx}[e^u] = e^u \cdot u'$$

Examples:

• $\frac{d}{dx}[e^{3x}] = e^{3x} \cdot 3 = 3e^{3x}$
• $\frac{d}{dx}[e^{x^2}] = e^{x^2} \cdot 2x = 2xe^{x^2}$
• $\frac{d}{dx}[e^{\sin x}] = e^{\sin x} \cdot \cos x$

General Exponential: $a^x$

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

Derivation: Write $a^x = e^{x\ln a}$, then apply the chain rule: $e^{x\ln a} \cdot \ln a = a^x \ln a$.

Examples:

• $\frac{d}{dx}[2^x] = 2^x \ln 2$
• $\frac{d}{dx}[10^x] = 10^x \ln 10$

The Natural Logarithm

$$\frac{d}{dx}[\ln x] = \frac{1}{x}, \quad x > 0$$

For $x \neq 0$: $\frac{d}{dx}[\ln|x|] = \frac{1}{x}$

With chain rule: $$\frac{d}{dx}[\ln(u)] = \frac{1}{u} \cdot u' = \frac{u'}{u}$$

Examples:

• $\frac{d}{dx}[\ln(x^2+1)] = \frac{2x}{x^2+1}$
• $\frac{d}{dx}[\ln(\sin x)] = \frac{\cos x}{\sin x} = \cot x$

General Logarithm: $\log_a x$

$$\frac{d}{dx}[\log_a x] = \frac{1}{x \ln a}$$

Derivation: Change of base: $\log_a x = \frac{\ln x}{\ln a}$, so the derivative is $\frac{1}{x\ln a}$.

Logarithmic Differentiation

Useful when:

• The exponent is a variable (not just a constant)
• The function is a complex product or quotient

Technique: Take $\ln$ of both sides, differentiate implicitly, solve for $y'$.

Example: Differentiate $y = x^x$.

Step 1: $\ln y = \ln(x^x) = x\ln x$

Step 2: Differentiate both sides (implicitly): $\frac{y'}{y} = \ln x + x \cdot \frac{1}{x} = \ln x + 1$

Step 3: Solve: $y' = y(\ln x + 1) = x^x(\ln x + 1)$

Differentiate $y = \frac{x^2\sqrt{x+1}}{(2x-3)^4}$