Derivatives of standard functions

IB Mathematics AI HL· IB AI HL Syllabus 2021 5.1· 15 min read

1. The Power Rule★☆☆☆☆⏱ 3 min

📘 Definition

Power Rule for Differentiation

For $f (x) = x^{n}$ , $f^{'} (x) = n x^{n - 1}$

The power rule gives the derivative of any power function with constant exponent $n$ , and works for all real values of $n$ (positive, negative, fractional, irrational).

Example:

For $f (x) = x^{3}$ , $f^{'} (x) = 3 x^{2}$

📐 Worked Example

Differentiate $f (x) = 5 x^{4} - \frac{2}{x ^{3}} + 7 x$ with respect to $x$ .

1
Rewrite all terms to match the power function form $x^{n}$ :
2
$f(x) = 5x^4 - 2x^{-3} + 7x^{1/2}$
3
Apply the power rule and constant multiple rule term-by-term:
4
First term: $\frac{d}{d x} (5 x^{4}) = 5 \times 4 x^{4 - 1} = 20 x^{3}$
5
Second term: $\frac{d}{d x} (- 2 x^{- 3}) = - 2 \times (- 3) x^{- 3 - 1} = 6 x^{- 4}$
6
Third term: $\frac{d}{d x} (7 x^{1/2}) = 7 \times \frac{1}{2} x^{(1/2 - 1)} = \frac{7}{2} x^{- 1/2}$
7
Rewrite back in root/reciprocal form for the final answer:
8
$f'(x) = 20x^3 + \frac{6}{x^4} + \frac{7}{2\sqrt{x}}$

Exam tip:

Always complete both steps of the power rule: multiply by the exponent, then subtract 1 from the exponent.

2. Exponential and Logarithmic Derivatives★★☆☆☆⏱ 4 min

📘 Definition

Derivatives of Exponential and Logarithmic Functions

For base $e$ (Euler's number): $\frac{d}{d x} e^{x} = e^{x}$ and $\frac{d}{d x} ln x = \frac{1}{x}$ . For any general positive base $a \neq = 1$ : $\frac{d}{d x} a^{x} = a^{x} ln a$ and $\frac{d}{d x} lo g_{a} x = \frac{1}{x l n a}$ .

📐 Worked Example

Find the derivative of $g (x) = 3 e^{x} - 2 ln (5 x) + 4^{x}$ .

1
Split the logarithm term using log rules to separate the constant: $ln (5 x) = ln 5 + ln x$
2
Differentiate term-by-term, noting that $ln 5$ is a constant (derivative = 0):
3
$\frac{d}{d x} (3 e^{x}) = 3 e^{x}$
4
$\frac{d}{d x} (- 2 ln 5 x) = - 2 (0 + \frac{1}{x}) = - \frac{2}{x}$
5
$\frac{d}{d x} (4^{x}) = 4^{x} ln 4$
6
Combine terms for the final answer:
7
$g'(x) = 3e^x - \frac{2}{x} + 4^x \ln 4$

3. Derivatives of Trigonometric Functions★★☆☆☆⏱ 4 min

📘 Definition

Derivatives of Basic Trigonometric Functions

All rules below only hold when $x$ is measured in radians. The derivatives are: $\frac{d}{d x} sin x = cos x$ , $\frac{d}{d x} cos x = - sin x$ , $\frac{d}{d x} tan x = sec^{2} x$ .

📐 Worked Example

Differentiate $h (x) = 4 tan x - 3 cos x + \frac{s i n x}{2}$ .

1
Apply the rules term-by-term:
2
$\frac{d}{d x} (4 tan x) = 4 sec^{2} x$
3
$\frac{d}{d x} (- 3 cos x) = - 3 (- sin x) = 3 sin x$
4
$\frac{d}{d x} (\frac{s i n x}{2}) = \frac{1}{2} cos x$
5
Final result:
6
$h'(x) = 4\sec^2 x + 3\sin x + \frac{1}{2}\cos x$

Exam tip:

Always convert any angle given in degrees to radians before differentiating trigonometric functions.

4. Derivatives of Inverse Trigonometric Functions★★★☆☆HL only⏱ 4 min

📘 Definition

Standard Inverse Trigonometric Derivatives

For IB AI HL, the required derivatives are: $\frac{d}{d x} arcsin x = \frac{1}{1 - x ^{2}}$ , $\frac{d}{d x} arccos x = - \frac{1}{1 - x ^{2}}$ , $\frac{d}{d x} arctan x = \frac{1}{1 + x ^{2}}$ .

📐 Worked Example

Find the derivative of $y = 5 arctan x + 2 arcsin x$ .

1
Apply the standard derivative rule to each term:
2
$\frac{d}{d x} (5 arctan x) = 5 \times \frac{1}{1 + x ^{2}} = \frac{5}{1 + x ^{2}}$
3
$\frac{d}{d x} (2 arcsin x) = 2 \times \frac{1}{1 - x ^{2}} = \frac{2}{1 - x ^{2}}$
4
Final derivative:
5
$\frac{dy}{dx} = \frac{5}{1+x^2} + \frac{2}{\sqrt{1-x^2}}$

Exam tip:

Don't forget the negative sign for the derivative of $arccos x$ — this is a common exam marking point.

5. Informal Limits and the Meaning of the Derivative★☆☆☆☆⏱ 5 min

Before applying derivative rules, it helps to understand what a derivative actually measures. In IB AI you treat limits informally: instead of an algebraic proof, you estimate the value a function approaches by reading off a table of values or looking at a graph. This intuition underpins two key ideas — the gradient of a tangent, and the rate of change.

📘 Definition

Informal limit

\lim_{x \to a} f(x)

The single value that f(x) gets closer and closer to as x gets closer to a (from both sides). You can estimate it numerically from a table of values near x = a, or read it from a graph, without any formal algebra.

Example:

As x approaches 2, values of \frac{x^2 - 4}{x - 2} get closer to 4, so the limit is 4.

📐 Worked Example

Estimate $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ using a table of values.

1
The expression is undefined at exactly x = 2, so evaluate it at x-values just below and just above 2:
2
$\begin{array}{c|ccccc} x & 1.9 & 1.99 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & 3.9 & 3.99 & 4.001 & 4.01 & 4.1 \end{array}$
3
From both sides the values close in on 4, so we estimate the limit is 4. (A GDC table does this instantly.)

The derivative is built from exactly this idea of a limit. The gradient of the curve at a point is estimated by the gradient of a chord joining two nearby points; as the second point slides towards the first, the chord's gradient approaches a single value — the gradient of the tangent line at that point. That limiting value is the derivative, and it also measures the instantaneous rate of change of the function.

📐 Worked Example

For $f (x) = x^{2}$ , estimate the gradient of the tangent at $x = 3$ by shrinking the interval of a chord.

1
The gradient of a chord from x = 3 to x = 3 + h is $\frac{f ( 3 + h ) - f ( 3 )}{h}$ . Compute it for smaller and smaller h:
2
$\begin{array}{c|ccc} h & 0.1 & 0.01 & 0.001 \\ \hline \text{chord gradient} & 6.1 & 6.01 & 6.001 \end{array}$
3
As h shrinks towards 0, the chord gradient approaches 6, so the tangent gradient at x = 3 is about 6.
4
This matches the power rule: $f^{'} (x) = 2 x$ gives $f^{'} (3) = 6$ . The rules you learned are shortcuts for this limiting process.

6. Common Pitfalls

Wrong move:

Forgetting negative signs when differentiating negative exponents, $cos x$ or $arccos x$ .

Why:

Sign errors are the most frequently marked error in basic differentiation questions.

Correct move:

Double check all negative exponents and trigonometric/inverse trigonometric terms for correct sign after differentiation.

Wrong move:

Differentiating $ln (k x)$ (constant $k$ ) as $\frac{1}{k x}$ .

Why:

Forgetting that $ln (k x) = ln k + ln x$ , and $ln k$ is a constant with derivative zero.

Correct move:

Split the logarithm to separate the constant term before differentiation, confirming the derivative is $\frac{1}{x}$ .

Wrong move:

Confusing power and exponential rules: differentiating $x^{n}$ (constant $n$ ) as $x^{n} ln x$ , or $a^{x}$ (constant $a$ ) as $x a^{x - 1}$ .

Why:

Mixing up power functions (variable base, constant exponent) and exponential functions (constant base, variable exponent).

Correct move:

Use power rule for $x^{n}$ (constant $n$ ) and exponential rule for $a^{x}$ (constant $a$ ).

Wrong move:

Differentiating a trigonometric function when $x$ is given in degrees.

Why:

All standard trigonometric derivative rules are derived for radians, not degrees.

Correct move:

Convert all angle measures to radians before applying derivative rules.

Wrong move:

Forgetting to subtract 1 from the exponent when applying the power rule, e.g. writing $\frac{d}{d x} (5 x^{3}) = 15 x^{3}$ .

Why:

Only completing the first step of the power rule (multiplying by the exponent) and missing the second step.

Correct move:

Always follow two steps for power rule: 1) multiply by the original exponent, 2) subtract 1 from the exponent.

7. Quick Reference Cheatsheet

Function Category	f(x)	f'(x)
Power	$x^{n}$ (n constant)	$n x^{n - 1}$
Exponential	$e^{x}$	$e^{x}$
Exponential	$a^{x}$ (a>0, a≠1)	$a^{x} ln a$
Logarithm	$ln x$	$1/ x$
Logarithm	$lo g_{a} x$	$1/ (x ln a)$
Trigonometric	$sin x$	$cos x$
Trigonometric	$cos x$	$- sin x$
Trigonometric	$tan x$	$sec^{2} x$
Inverse Trig (HL)	$arcsin x$	$1/ 1 - x^{2}$
Inverse Trig (HL)	$arccos x$	$- 1/ 1 - x^{2}$
Inverse Trig (HL)	$arctan x$	$1/ (1 + x^{2})$

What's Next

Mastering these standard derivative rules is the foundation for all remaining calculus work in IB AI HL. Next, you will learn how to combine these base rules to differentiate more complex functions created by composition, multiplication, and division of standard functions. These combined rules are used for every calculus application on the exam, from finding tangents and optimisation to integration and solving differential equations, so getting comfortable with these standard rules now will prevent mistakes later in the course.