Differentiation of trigonometric, exponential and logarithmic functions

IB Mathematics Analysis and Approaches SL· Unit 5: Calculus· 20 min read

1. Differentiation of Trigonometric Functions★★☆☆☆⏱ 7 min

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📘 Definition

Standard derivatives of basic trigonometric functions

For $x$ measured in radians (standard for calculus), the derivatives are: $\frac{d}{d x} (sin x) = cos x$ , $\frac{d}{d x} (cos x) = - sin x$ , $\frac{d}{d x} (tan x) = sec^{2} x$ .

Example:

If $f (x) = 3 sin x$ , then $f^{'} (x) = 3 cos x$ .

To differentiate composite trigonometric functions (functions of the form $f (g (x))$ where $f$ is trigonometric), you combine these standard derivatives with the chain rule.

📐 Worked Example

Find the derivative of $f (x) = tan (3 x^{2} + 2 x)$

1
Split into outer and inner functions for the chain rule: outer $f (u) = tan u$ , inner $u = 3 x^{2} + 2 x$
2
Calculate derivatives of each part:
3
$\frac{df}{du} = \sec^2 u, \quad \frac{du}{dx} = 6x + 2$
4
Apply the chain rule $\frac{df}{d x} = \frac{df}{d u} \times \frac{d u}{d x}$ , then substitute back $u$ :
5
$f'(x) = (6x + 2)\sec^2(3x^2 + 2x)$

Exam tip:

Always check you have applied the chain rule to composite trigonometric functions; forgetting the inner function derivative is the most common mistake here.

2. Differentiation of Exponential Functions★★☆☆☆⏱ 6 min

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📘 Definition

Derivatives of exponential functions

For the natural exponential: $\frac{d}{d x} (e^{x}) = e^{x}$ . For general exponential $a^{x}$ ( $a > 0, a \neq = 1$ ): $\frac{d}{d x} (a^{x}) = a^{x} ln a$ .

Example:

$\frac{d}{d x} (5^{x}) = 5^{x} ln 5$

For composite exponentials of the form $e^{g (x)}$ , the derivative simplifies to $g^{'} (x) e^{g (x)}$ by the chain rule. This form appears very frequently in IB problems.

📐 Worked Example

Find the derivative of $y = x^{2} e^{3 x}$ and evaluate the gradient at $x = 0$

1
This is a product of two functions, so use the product rule $y^{'} = u^{'} v + u v^{'}$ , with $u = x^{2}$ , $v = e^{3 x}$
2
Calculate derivatives of each part:
3
$u' = 2x, \quad v' = 3e^{3x}$
4
Substitute into the product rule and factor:
5
$y' = 2x(e^{3x}) + x^2(3e^{3x}) = e^{3x}(2x + 3x^2)$
6
Evaluate at $x = 0$ :
7
$y'(0) = e^{0}(0 + 0) = 1 \times 0 = 0$

Exam tip:

Remember that $e^{x}$ is its own derivative, do not apply the power rule as you would for $x^{n}$ .

3. Differentiation of Logarithmic Functions★★★☆☆⏱ 7 min

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📘 Definition

Derivatives of logarithmic functions

For natural logarithm ( $x > 0$ ): $\frac{d}{d x} (ln x) = \frac{1}{x}$ . For general logarithm ( $a > 0, a \neq = 1, x > 0$ ): $\frac{d}{d x} (lo g_{a} x) = \frac{1}{x l n a}$ .

Example:

$\frac{d}{d x} (lo g_{10} x) = \frac{1}{x l n 10}$

For composite logarithms of the form $ln (g (x))$ , apply the chain rule to get the simple result $\frac{g ^{'} ( x )}{g ( x )}$ . This identity is used for logarithmic differentiation of complex products and quotients.

📐 Worked Example

Differentiate $f (x) = ln (x^{3} - 2 x + 1)$ , stating the domain where the derivative is valid.

1
Use the chain rule result for $ln (g (x))$ , where $g (x) = x^{3} - 2 x + 1$
2
First calculate the derivative of the inner function:
3
$g'(x) = 3x^2 - 2$
4
Substitute into the derivative formula for composite logs:
5
$f'(x) = \frac{3x^2 - 2}{x^3 - 2x + 1}$
6
The derivative is only valid where the original function is defined, so for $x^{3} - 2 x + 1 > 0$ .

Exam tip:

Remember that the derivative of $ln (g (x))$ is always $\frac{g ^{'} ( x )}{g ( x )}$ this result will save you time on exam day.

4. Common Pitfalls

Wrong move:

Writing $\frac{d}{d x} (sin x) = - cos x$ and $\frac{d}{d x} (cos x) = sin x$ , swapping the signs.

Why:

The signs of the derivatives of sine and cosine are commonly mixed up during memorization.

Correct move:

Memorize the correct derivatives: $\frac{d}{d x} (sin x) = cos x$ , $\frac{d}{d x} (cos x) = - sin x$ .

Wrong move:

Forgetting the chain rule for $e^{2 x}$ , writing $\frac{d}{d x} (e^{2 x}) = e^{2 x}$ .

Why:

The inner function $2 x$ has a derivative of 2 that must be included via the chain rule.

Correct move:

Apply the chain rule to get $\frac{d}{d x} (e^{2 x}) = 2 e^{2 x}$ .

Wrong move:

Confusing derivatives of $e^{x}$ and $ln x$ , writing $\frac{d}{d x} (ln x) = e^{x}$ or $\frac{d}{d x} (e^{x}) = \frac{1}{x}$ .

Why:

These two core functions have distinct derivatives that are often swapped by students.

Correct move:

Remember: $\frac{d}{d x} (e^{x}) = e^{x}$ and $\frac{d}{d x} (ln x) = \frac{1}{x}$ .

Wrong move:

Applying the power rule to $a^{x}$ , writing $\frac{d}{d x} (a^{x}) = x a^{x - 1}$ .

Why:

The power rule applies to functions with variable base and constant exponent, not constant base and variable exponent.

Correct move:

Use the exponential derivative rule: $\frac{d}{d x} (a^{x}) = a^{x} ln a$ .

5. Quick Reference Cheatsheet

Function	Derivative
$sin x$	$cos x$
$cos x$	$- sin x$
$tan x$	$sec^{2} x$
$e^{x}$	$e^{x}$
$a^{x}$	$a^{x} ln a$
$ln x$	$\frac{1}{x}$
$lo g_{a} x$	$\frac{1}{x l n a}$
$ln (g (x))$	$\frac{g ^{'} ( x )}{g ( x )}$
$e^{g (x)}$	$g^{'} (x) e^{g (x)}$

6. Frequently Asked

Are these derivatives given in the formula booklet?

No, these standard derivatives are not provided in the IB AA SL formula booklet, so you must memorize them for the exam.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Differentiate composite trigonometric function
2022 · 2
Find gradient of exponential curve at point
2023 · 1
Differentiate product of log and trig

What's Next

Now that you have mastered these standard derivatives, you can apply them to a wide range of calculus problems, from finding gradients and tangents to curves to solving optimization problems and differential equations. These derivatives are foundational for all further work in IB AA SL calculus, so it is critical that you memorize them and can apply them fluently in combination with the chain, product and quotient rules. Next, you will move on to using these derivatives to solve applied problems involving tangents and normals, then to further applications like finding stationary points and optimization problems. Mastery of this subtopic will make all subsequent calculus topics much easier, as every advanced calculus concept in IB AA SL builds on these core derivative rules.