Derivatives of Exponential and Logarithm Functions for MATH 122

Exam Relevance for MATH 122

Likelihood of appearing: High

Exponential and log derivatives are heavily tested in MATH 122 due to business applications like compound interest.

Lesson

Derivatives of Exponential Functions

The Natural Exponential: $e^x$

The function $e^x$ is special because it is its own derivative:

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

This is the only function (other than 0) with this property!

Example 1: Basic Exponential

Problem: Find $\frac{d}{dx}[e^x]$

Step 1: Apply the rule $\frac{d}{dx}[e^x] = e^x$:

$$\boxed{e^x}$$

Example 2: Exponential with Chain Rule

Problem: Find $\frac{d}{dx}[e^{3x}]$

Step 1: Identify this as $e^{f(x)}$ where $f(x) = 3x$

Step 2: Apply the chain rule: $\frac{d}{dx}[e^{f(x)}] = e^{f(x)} \cdot f'(x)$

$$\frac{d}{dx}[e^{3x}] = e^{3x} \cdot 3 = \boxed{3e^{3x}}$$

Example 3: More Complex Exponential

Problem: Find $\frac{d}{dx}[e^{x^2}]$

Step 1: Here $f(x) = x^2$, so $f'(x) = 2x$

Step 2: Apply chain rule: $$\frac{d}{dx}[e^{x^2}] = e^{x^2} \cdot 2x = \boxed{2xe^{x^2}}$$

General Exponential: $a^x$

For any positive base $a \neq 1$:

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

Why the $\ln(a)$? We can rewrite $a^x = e^{x\ln(a)}$, then use chain rule.

Example 4: General Base

Problem: Find $\frac{d}{dx}[2^x]$

Step 1: Apply the rule $\frac{d}{dx}[a^x] = a^x \ln(a)$ with $a = 2$: $$\frac{d}{dx}[2^x] = 2^x \cdot \ln(2)$$

$$\boxed{2^x \ln 2}$$

Derivatives of Logarithmic Functions

The Natural Logarithm: $\ln x$

$$\frac{d}{dx}[\ln x] = \frac{1}{x} \quad \text{(for } x > 0 \text{)}$$

Example 5: Basic Logarithm

Problem: Find $\frac{d}{dx}[\ln x]$

Step 1: Apply the rule $\frac{d}{dx}[\ln x] = \frac{1}{x}$:

$$\boxed{\frac{1}{x}}$$

Example 6: Logarithm with Chain Rule

Problem: Find $\frac{d}{dx}[\ln(3x+1)]$

Step 1: This is $\ln(f(x))$ where $f(x) = 3x+1$

Step 2: Apply chain rule: $\frac{d}{dx}[\ln f(x)] = \frac{f'(x)}{f(x)}$

$$\frac{d}{dx}[\ln(3x+1)] = \frac{3}{3x+1} = \boxed{\frac{3}{3x+1}}$$

Example 7: Logarithm of a Product

Problem: Find $\frac{d}{dx}[\ln(x^2)]$

Solution Method 1: Use chain rule with $f(x) = x^2$: $$\frac{d}{dx}[\ln(x^2)] = \frac{2x}{x^2} = \frac{2}{x}$$

Method 2: Simplify first using log properties: $\ln(x^2) = 2\ln x$ $$\frac{d}{dx}[2\ln x] = 2 \cdot \frac{1}{x} = \frac{2}{x}$$

$$\boxed{\frac{2}{x}}$$

General Logarithm: $\log_a x$

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

Example 8: Log Base 10

Problem: Find $\frac{d}{dx}[\log_{10} x]$

Step 1: Apply the rule $\frac{d}{dx}[\log_a x] = \frac{1}{x \ln(a)}$ with $a = 10$: $$\frac{d}{dx}[\log_{10} x] = \frac{1}{x \ln(10)}$$

$$\boxed{\frac{1}{x \ln 10}}$$

Summary Table

Function	Derivative
$e^x$	$e^x$
$e^{f(x)}$	$e^{f(x)} \cdot f'(x)$
$a^x$	$a^x \ln(a)$
$\ln x$	$\frac{1}{x}$
$\ln f(x)$	$\frac{f'(x)}{f(x)}$
$\log_a x$	$\frac{1}{x \ln(a)}$

Common Mistakes and Misunderstandings

❌ Mistake: Forgetting the chain rule with exponentials

Wrong: $\frac{d}{dx}[e^{5x}] = e^{5x}$

Why it's wrong: The exponent $5x$ is not just $x$, so you need the chain rule.

Correct: $\frac{d}{dx}[e^{5x}] = e^{5x} \cdot 5 = 5e^{5x}$

❌ Mistake: Confusing $\frac{d}{dx}[e^x]$ with $\frac{d}{dx}[x^e]$

$\frac{d}{dx}[e^x] = e^x$ (exponential rule — base is constant)
$\frac{d}{dx}[x^e] = ex^{e-1}$ (power rule — exponent is constant)

The rules depend on what's constant and what's variable!

Formulas & Reference

Derivative of e^x

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

The exponential function e^x is its own derivative

Variables:

$e$:: Euler's number (≈ 2.718)
$e^x$:: the natural exponential function

Derivative of e^(f(x))

\frac{d}{dx}[e^{f(x)}] = e^{f(x)} \cdot f'(x)

Chain rule applied to exponential functions

Variables:

$f(x)$:: the exponent function
$f'(x)$:: derivative of the exponent

Derivative of a^x

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

Derivative of exponential with any base a

Variables:

$a$:: the base (a > 0, a ≠ 1)
$ln(a)$:: natural logarithm of the base

Derivative of ln(x)

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

Derivative of the natural logarithm

Variables:

$x$:: must be positive (x > 0)

Derivative of ln(f(x))

\frac{d}{dx}[\ln f(x)] = \frac{f'(x)}{f(x)}

Chain rule applied to natural logarithm

Variables:

$f(x)$:: the inner function (must be positive)
$f'(x)$:: derivative of the inner function

Derivative of log_a(x)

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

Derivative of logarithm with any base

Variables:

$a$:: the base of the logarithm
$ln(a)$:: natural log of the base

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