2.7 - Derivatives of cos x, sin x, e^x, and ln x

Introduction

These are the derivative rules for sine, cosine, exponential, and logarithmic functions. You’ll use them constantly, so memorizing them isn’t optional.

Derivatives of Trig Functions

\frac{d}{dx}[\sin x] = \cos x

The derivative of sine is cosine.

\frac{d}{dx}[\cos x] = -\sin x

The derivative of cosine is negative sine.

Memory tip: the derivative of cosine picks up a negative sign. Functions that start with “c” (cosine, and later cotangent and cosecant) all have a negative sign in front of their derivatives.

Example 1: Basic Trig Derivatives

Find the derivative of

f(x) = 3\sin x - 4\cos x

f'(x) = 3\cos x - 4(-\sin x) = 3\cos x + 4\sin x

The

-4\cos x

becomes

-4(-\sin x)

, and the two negatives cancel into a positive.

Example 2: Mixed Functions

Differentiate

g(x) = 5\cos x + 2x^3 - \sin x

g'(x) = 5(-\sin x) + 6x^2 - \cos x = -5\sin x + 6x^2 - \cos x

Properties Worth Remembering First

Natural Logarithm Properties

\ln(1) = 0 \qquad \ln(0) = \text{undefined}

Why is

\ln(0)

undefined? Because the graph of

y = \ln(x)

has a vertical asymptote at

x = 0

2.7 - Derivatives of cos x, sin x, e^x, and ln x

Since the function never actually reaches

x = 0

\ln(0)

is undefined.

Exponential Properties

e^0 = 1

(Any nonzero number to the zero power is 1.)

Inverse Properties

e^{\ln(a)} = a \qquad \ln(e^a) = a

The natural exponential and the natural log are inverses, so they undo each other.

Derivatives of Exponential Functions

Rule 1: Derivative of $e^x$

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

The derivative of

e^x

is itself. One of the most elegant results in calculus.

Rule 2: Derivative of $a^x$ (general base)

For any positive constant

a \neq 1

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

The pattern: keep the function, then multiply by the natural log of the base.

Why $e^x$ is special: when

a = e

\frac{d}{dx}[e^x] = e^x \cdot \ln(e) = e^x \cdot 1 = e^x

Since

\ln(e) = 1

, the extra factor just disappears.

Example 3: Exponential with Base $e$

Find the derivative of

f(x) = 7e^x + 3x^2

f'(x) = 7e^x + 6x

The derivative of

e^x

is just

e^x

, so the 7 stays put.

Example 4: Exponential with Another Base

Differentiate

h(x) = 5^x - 2x

h'(x) = 5^x \ln(5) - 2

Since the base is 5 and not

e

, you have to include

\ln(5)

Example 5: Multiple Exponentials

Find

g'(x)

for

g(x) = 4e^x + 2^x

g'(x) = 4e^x + 2^x \ln(2)

The $4e^x$ term gives $4e^x$

The $2^x$ term gives $2^x \ln(2)$

Derivatives of Logarithmic Functions

Rule 1: Natural Log

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

Simple and important.

This rule only works for

x > 0

, because that’s the domain of

\ln x

. The formula

\frac{1}{x}

gives a number for negative

x

too, but

\ln x

doesn’t exist there, so neither does its derivative. (Remember from Topic 2.4: a point that isn’t in the domain of

f

isn’t in the domain of

f'

Rule 2: General Log

For any positive constant

a \neq 1

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

The pattern:

\frac{1}{x}

multiplied by

\frac{1}{\ln(\text{base})}

Why natural log is special: when

a = e

\frac{d}{dx}[\ln x] = \frac{1}{x \ln(e)} = \frac{1}{x \cdot 1} = \frac{1}{x}

Example 6: Natural Log

Find the derivative of

f(x) = 3\ln x + 5x

f'(x) = 3 \cdot \frac{1}{x} + 5 = \frac{3}{x} + 5

Example 7: Log Base 10

Differentiate

g(x) = \log_{10} x + 4x^2

g'(x) = \frac{1}{x \ln(10)} + 8x

Base 10 isn’t

e

, so

\ln(10)

goes in the denominator.

Example 8: Log with Another Base

Find

h'(x)

for

h(x) = \log_3 x - 2\cos x

h'(x) = \frac{1}{x \ln(3)} - 2(-\sin x) = \frac{1}{x \ln(3)} + 2\sin x

Combining Multiple Rules

Example 9: Mixed Exponential and Trig

Find the derivative of

f(x) = 4\sin x + 6e^x

f'(x) = 4\cos x + 6e^x

Example 10: All Function Types

Differentiate

g(x) = 3^x - 5\cos x + e^x

g'(x) = 3^x \ln(3) - 5(-\sin x) + e^x = 3^x \ln(3) + 5\sin x + e^x

Example 11: Log and Trig

Find

h'(x)

for

h(x) = 2\log_5 x + \dfrac{\sin x}{3}

Rewrite:

h(x) = 2\log_5 x + \frac{1}{3}\sin x

h'(x) = \frac{2}{x \ln(5)} + \frac{1}{3}\cos x

Evaluating Derivatives at Specific Points

Example 12: Evaluate at a Point

Given

f(x) = 2\ln x + e^x

, find

f'(3)

Step 1: Find the derivative.

f'(x) = \frac{2}{x} + e^x

Step 2: Plug in

x = 3

f'(3) = \frac{2}{3} + e^3

That’s the exact answer. Don’t try to simplify

e^3

any further, it’s irrational (about 20.086).

Example 13: Trig Function at a Specific Point

Find

g'(\pi)

g(x) = 4\cos x + \dfrac{\sin x}{2}

Step 1: Find the derivative.

g'(x) = -4\sin x + \frac{1}{2}\cos x

Step 2: Evaluate at

x = \pi

g'(\pi) = -4\sin(\pi) + \frac{1}{2}\cos(\pi)

Step 3: Use unit circle values.

\sin(\pi) = 0

and

\cos(\pi) = -1

g'(\pi) = -4(0) + \frac{1}{2}(-1) = -\frac{1}{2}

Example 14: Another Trig Evaluation

Find

h'\left(\frac{\pi}{2}\right)

for

h(x) = 3\sin x - 2\cos x

h'(x) = 3\cos x + 2\sin x

Using the unit circle,

\cos\left(\frac{\pi}{2}\right) = 0

and

\sin\left(\frac{\pi}{2}\right) = 1

h'\left(\frac{\pi}{2}\right) = 3(0) + 2(1) = 2

Using Derivatives to Evaluate Limits

In Topic 2.1 you learned to spot the function and the point hiding in a limit like

\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

. Now that you know these derivative rules, you can go one step further and actually evaluate the limit, because it equals

f'(a)

. Many of these limits give

\frac{0}{0}

and are hard to do with algebra, but they take one line with a derivative rule.

Steps:

Match the limit to $\frac{f(a + h) - f(a)}{h}$ or $\frac{f(x) - f(a)}{x - a}$ .

Name the function $f$ and the point $a$ .

Find $f'(x)$ with a derivative rule.

Plug in $a$ . The limit equals $f'(a)$ .

Example 15: A Sine Limit

Evaluate:

\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{6} + h\right) - \frac{1}{2}}{h}

Step 1: Since

\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}

, this is

\frac{f(a + h) - f(a)}{h}

with

f(x) = \sin x

and

a = \frac{\pi}{6}

Step 2: The derivative of

\sin x

\cos x

, so the limit is

f'\left(\frac{\pi}{6}\right)

\lim_{h \to 0} \frac{\sin\left(\frac{\pi}{6} + h\right) - \frac{1}{2}}{h} = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}

Example 16: A Natural Log Limit

Evaluate

\lim_{x \to 1} \frac{\ln x}{x - 1}

Plugging in

x = 1

gives

\frac{0}{0}

. But

\ln(1) = 0

, so the numerator is really

\ln x - \ln(1)

\lim_{x \to 1} \frac{\ln x - \ln(1)}{x - 1}

That’s

\frac{f(x) - f(a)}{x - a}

with

f(x) = \ln x

and

a = 1

. Since

f'(x) = \frac{1}{x}

, the limit is

f'(1) = \frac{1}{1} = 1

Example 17: A Cosine Limit

Evaluate

\lim_{h \to 0} \frac{\cos h - 1}{h}

Since

\cos(0) = 1

, write it as

\frac{\cos(0 + h) - \cos(0)}{h}

. That’s

\frac{f(a + h) - f(a)}{h}

with

f(x) = \cos x

and

a = 0

\lim_{h \to 0} \frac{\cos h - 1}{h} = -\sin(0) = 0

Key idea: the number being subtracted in the numerator has to equal

f(a)

. Check that before you use the derivative.

Practice Problems

Problem 1. Find the derivative:

f(x) = \sin x + \cos x + x^2

$f'(x) = \cos x - \sin x + 2x$

Problem 2. Differentiate:

g(x) = e^x + 4^x

$g'(x) = e^x + 4^x \ln(4)$

Problem 3. Find

h'(x)

for

h(x) = \ln x + \log_2 x

$h'(x) = \frac{1}{x} + \frac{1}{x \ln(2)}$

Problem 4. Evaluate

f'(1)

f(x) = 5\ln x + 3e^x

$f'(x) = \frac{5}{x} + 3e^x$ , so $f'(1) = 5 + 3e$

Problem 5. Find

g'(0)

for

g(x) = \sin x + \cos x

$g'(x) = \cos x - \sin x$ , so $g'(0) = 1 - 0 = 1$

Problem 6. Evaluate

\lim_{h \to 0} \frac{e^{3 + h} - e^3}{h}

This is the derivative of $e^x$ at $x = 3$ , so the limit is $e^3$ .

Summary of Derivative Rules

Most Common (memorize these)

Function	Derivative
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$e^x$	$e^x$
$\ln x$	$\frac{1}{x}$

Less Common (still important)

Function	Derivative
$a^x$	$a^x \ln(a)$
$\log_a x$	$\frac{1}{x \ln(a)}$

Key Takeaways

Derivative of sine is cosine (positive)

Derivative of cosine is negative sine

Derivative of $e^x$ is itself

Derivative of $\ln x$ is $\frac{1}{x}$

For other bases ( $a^x$ or $\log_a x$ ), you have to include $\ln(a)$

$e^x$ and $\ln x$ are both special because $\ln(e) = 1$

Remember the inverse properties: $e^{\ln(a)} = a$ and $\ln(e^a) = a$

When evaluating at specific angles, use unit circle values

These rules combine with the power rule, constant multiple, and sum/difference rules

A limit that matches the definition of the derivative equals $f'(a)$ , so you can evaluate it with a derivative rule

Back Next

2.7 - Derivatives of cos x, sin x, e^x, and ln x

Introduction

Derivatives of Trig Functions

Example 1: Basic Trig Derivatives

Example 2: Mixed Functions

Properties Worth Remembering First

Natural Logarithm Properties

Exponential Properties

Inverse Properties

Derivatives of Exponential Functions

Rule 1: Derivative of exe^xex

Rule 2: Derivative of axa^xax (general base)

Example 3: Exponential with Base eee

Example 4: Exponential with Another Base

Example 5: Multiple Exponentials

Derivatives of Logarithmic Functions

Rule 1: Natural Log

Rule 2: General Log

Example 6: Natural Log

Example 7: Log Base 10

Example 8: Log with Another Base

Combining Multiple Rules

Example 9: Mixed Exponential and Trig

Example 10: All Function Types

Example 11: Log and Trig

Evaluating Derivatives at Specific Points

Example 12: Evaluate at a Point

Example 13: Trig Function at a Specific Point

Example 14: Another Trig Evaluation

Using Derivatives to Evaluate Limits

Example 15: A Sine Limit

Example 16: A Natural Log Limit

Example 17: A Cosine Limit

Practice Problems

Summary of Derivative Rules

Most Common (memorize these)

Less Common (still important)

Key Takeaways

Rule 1: Derivative of $e^x$

Rule 2: Derivative of $a^x$ (general base)

Example 3: Exponential with Base $e$