Differentiate powers, trigonometric, exponential, and logarithmic functions

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Calculus

AA SL 5.3: derivative rules, meaning and applications

Differentiation describes how quickly a function changes. For a graph, the derivative at a point is the gradient of the tangent there. In context, it can represent a rate such as change in height per second. This lesson focuses on rules for powers, trigonometric, exponential and logarithmic functions. You should be comfortable with powers, function notation, basic algebra and the fact that trigonometric derivatives use angles measured in radians.

What you will learn

Interpret a derivative as a gradient that may vary from point to point.
Differentiate powers, sine, cosine, exponential functions and logarithms using standard rules.
Use the chain rule for a function with a simple inner expression.
Connect a derivative rule to a graph, a numerical estimate and a contextual rate.

1. From gradients to derivative notation

For a straight line, the gradient is constant. For a curve, the gradient usually changes with the input. The derivative gives the gradient at a particular input. If the function is written as

y=f(x)

, its derivative may be written as

f'(x)

or

\frac{dy}{dx}

. The derivative is itself a function: substituting a particular

x

gives the gradient at that point.

A numerical estimate of the gradient near

x=a

can be made by choosing a nearby value

h

and calculating the change in output divided by the change in input:

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

. Smaller values of

h

generally give a closer estimate, though calculator rounding can affect the result. Differentiation rules give the exact derivative, rather than an estimate.

A derivative has units. If

f(x)

is a distance in metres and

x

is time in seconds, then

f'(x)

is measured in metres per second. The graph of f' can also be useful: positive values indicate that

f

is increasing, and negative values indicate that it is decreasing.

f'(x)=\frac{dy}{dx}

The derivative at an input is the tangent gradient there.
A derivative can be interpreted as a rate of change, with units.
A numerical gradient is an estimate; a differentiation rule gives an exact result.

2. The core differentiation rules

For a power of

x

, multiply by the exponent and reduce the exponent by one. This applies to positive, zero and negative integer powers wherever the original function is defined. For example, the rule for

x^5

gives

5x^4

. A constant has derivative zero, and a constant multiple can be kept in front of the derivative.

Trigonometric differentiation uses radians. The derivative of sine is cosine, and the derivative of cosine is negative sine. The derivative of tangent is the reciprocal of cosine squared, at inputs where tangent is defined. These rules describe how the graph’s gradient changes with its angle input; using degrees would require an additional conversion factor.

For the natural exponential function

e^x

, the derivative is unchanged. For a general exponential function

a^x

, where

a>0

and

a\ne1

, the derivative is

a^x\ln a

. The logarithm is the natural logarithm. Thus the base matters: not every exponential function has itself as its derivative.

The natural logarithm has derivative

1/x

for

x>0

. A logarithm with base

a

can be written as

\frac{\ln x}{\ln a}

, so its derivative is

\frac{1}{x\ln a}

, again for

x>0

. The domain restriction is important: the real-valued logarithm is not defined for zero or negative inputs.

When a rule is applied to an inner expression, use the chain rule: differentiate the outer function with respect to its input, then multiply by the derivative of the inner expression. For example, differentiating

\sin(3x)

gives

\cos(3x)

multiplied by

3

. The chain rule also applies to powers, exponentials and logarithms.

\begin{aligned}\frac{d}{dx}(x^n)&=nx^{n-1},&\frac{d}{dx}(\sin x)&=\cos x,\\\frac{d}{dx}(\cos x)&=-\sin x,&\frac{d}{dx}(e^x)&=e^x,\\\frac{d}{dx}(a^x)&=a^x\ln a,&\frac{d}{dx}(\ln x)&=\frac{1}{x}\end{aligned}

The power rule is $x^n\mapsto nx^{n-1}$ .
For trigonometric derivatives, use radians.
For $a^x$ , include the factor $\ln a$ .
For logarithms, state the domain and differentiate any inner expression using the chain rule.

3. Reading derivatives in different ways

Symbolically, differentiation turns a rule for

f(x)

into a rule for

f'(x)

. Graphically, the value

f'(a)

is the gradient of the tangent to the graph of

f

at

x=a

. If

f'(a)=0

, the tangent is horizontal; this alone does not tell you whether the point is a maximum, a minimum or neither.

Numerically, a graphing calculator can display a tangent or estimate the derivative at a chosen input. Use that output as a check: first find the derivative by a rule, then compare the calculator’s gradient with the value of

f'(a)

. Check that the input is in the function’s domain and that angle settings are in radians for trigonometric functions.

In a context, explain what the derivative measures and include units. If a height model is measured in metres and time in seconds, the derivative is a vertical rate in metres per second. The sign tells the direction of change relative to the chosen positive direction.

f'(a)=\text{gradient at }x=a

The derivative graph records the gradients of the original graph.
Calculator gradients are checks, not substitutes for showing the differentiation rule.
A contextual derivative needs an interpretation and appropriate units.

4. A reliable method for differentiation questions

First identify the function type and any inner expression. Apply the matching rule and, if needed, the chain rule. Simplify the result only as far as useful. Then check the domain: a logarithm may restrict the input, and tangent is undefined where cosine is zero. Finally, substitute a requested input and interpret the value with the correct units.

For an exam-style response, make the reasoning visible. Name the relevant rule in words, show the derivative, and state any requested value or interpretation. If using technology to check a result, give the analytical derivative as well; a calculator display alone does not show why the answer is correct.

\frac{d}{dx}f(g(x))=f'(g(x))g'(x)

Identify the outer and inner functions before differentiating a composite expression.
Check domains and radians where relevant.
Keep exact results when possible and round only when requested.

Worked example

A power combined with a trigonometric function

Differentiate

f(x)=4x^3-2\sin x

and find the gradient at

x=0

.

Differentiate each term
Use the power rule on the first term and the sine rule on the second. The constant multiplier remains in front.
$f'(x)=12x^2-2\cos x$
Evaluate at the input
Substitute $x=0$ . Since $\cos 0=1$ , the gradient is negative two.
$f'(0)=12(0)^2-2\cos 0=-2$

Answer: The derivative is

f'(x)=12x^2-2\cos x

, and the gradient at

x=0

is

-2

.

Check: A graphing calculator in radian mode should show a tangent gradient of approximately

-2

at

x=0

. The derivative rule gives the exact value.

Worked example

An exponential with an inner expression

Differentiate

g(x)=5^{2x-1}

and find

g'(1)

to three significant figures.

Apply the exponential rule
For $a^{u}$ , differentiate with respect to the inner input to obtain $a^u\ln a$ , then multiply by the derivative of $u$ . Here the inner expression is $2x-1$ , whose derivative is $2$ .
$g'(x)=5^{2x-1}\ln 5\cdot 2$
Substitute and evaluate
At $x=1$ , the exponent is $1$ . Evaluating $10\ln 5$ gives approximately $16.1$ to three significant figures.
$g'(1)=10\ln 5\approx16.1$

Answer: The derivative is

g'(x)=2\cdot5^{2x-1}\ln 5

, and

g'(1)\approx16.1

to three significant figures.

Check: A calculator derivative estimate near

x=1

should be close to

16.1

. The factor

2

is required because the exponent changes at rate

2

.

Worked example

A logarithm and a contextual rate

A model gives

H(t)=3\ln(t+2)

metres, where

t

is time in seconds and

t\geq0

. Find the rate of change at

t=2

.

Differentiate the model
The outer logarithm has derivative equal to one over its input. Multiply by the derivative of the inner expression $t+2$ , which is $1$ ; retain the multiplier $3$ .
$H'(t)=\frac{3}{t+2}$
Evaluate and include units
Substitute $t=2$ . The result is a positive rate, so the modelled height is increasing at that instant.
$H'(2)=\frac{3}{4}=0.75\text{ m s}^{-1}$

Answer: At

t=2

seconds, the height is increasing at

0.75

metres per second.

Check: The model is defined for

t\geq0

, so

t=2

is allowed. The derivative’s units are metres per second because height is measured in metres and time in seconds.

Common mistakes and how to avoid them

Reducing the power without multiplying by the original exponent.

Correction: For

x^n

, multiply by

n

and then reduce the exponent by one.

Writing the derivative of

a^x

as just

a^x

for every base.

Correction: Include the factor

\ln a

unless the base is

e

.

Forgetting to multiply by the derivative of an inner expression.

Correction: Apply the chain rule: differentiate the outer function, then multiply by the inner derivative.

Using trigonometric derivative rules with degree-mode angles.

Correction: These rules assume angles are measured in radians; set calculator angle mode to radians when checking.

Giving a logarithm derivative without considering its domain.

Correction: For

\ln u

, require

u>0

in the real-valued setting.

Lesson summary

A derivative gives the gradient of a tangent and can represent a rate of change.
Use the power, trigonometric, exponential and logarithmic rules appropriate to the function.
For a composite function, multiply by the derivative of its inner expression.
Interpret calculator checks carefully, use radians for trigonometric differentiation, and include units in contextual answers.

Check your understanding

Question 1

What is the derivative of

x^4

?

$4x^3$
$x^3$
$4x^4$
$x^5$

Show answer and explanation

4x^3

The power rule multiplies by the exponent and reduces the exponent by one.

Question 2

What is the derivative of

3^x

?

$3^x$
$3^x\ln 3$
$x3^{x-1}$
$\frac{1}{x\ln 3}$

Show answer and explanation

3^x\ln 3

The derivative of

a^x

is

a^x\ln a

, so the base

3

contributes the factor

\ln 3

.

Question 3

For

y=\cos x

, what is the gradient at

x=0

?

$1$
$0$
$-1$
It is undefined.

Show answer and explanation

0

The derivative is

-\sin x

. At

x=0

, this is

0

.

Key terms

Derivative: A function giving the gradient of a curve at each input, or the corresponding rate of change.
Tangent: A straight line that touches a curve at a point and has the curve’s gradient there.
Chain rule: A rule for differentiating a function of an inner expression: differentiate the outer function and multiply by the inner derivative.
Natural logarithm: The logarithm with base $e$ , written $\ln x$ for positive $x$ .

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