# Differentiation of trigonometric, exponential and logarithmic functions

> IB Mathematics Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u5-differentiation-of-trigonometric-exponential-and/

This subtopic introduces the standard derivatives of core trigonometric, exponential and logarithmic functions. You will learn to combine these with existing differentiation rules to solve a wide range of calculus problems common in IB exams.

**Prerequisites:** [Basic differentiation rules](https://www.owlsprep.com/study/ib-math-aa-sl-u5-basic-differentiation-rules/); [Chain, product and quotient rules](https://www.owlsprep.com/study/ib-math-aa-sl-u5-chain-product-quotient-rules/); [Exponential and logarithmic properties](https://www.owlsprep.com/study/ib-math-aa-sl-u2-exponential-logarithmic-functions/)

## Learning objectives

- Recall and apply standard derivative rules for sine, cosine and tangent functions
- Differentiate natural exponential and general exponential functions
- Differentiate natural logarithmic and general logarithmic functions
- Combine these rules with chain, product and quotient rules for composite functions

## Differentiation of Trigonometric Functions

**Standard derivatives of basic trigonometric functions** — For $x$ measured in radians (standard for calculus), the derivatives are: $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$, $\frac{d}{dx}(\tan x) = \sec^2 x$.

*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 $f(x) = \tan(3x^2 + 2x)$

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

*Calculator:* forbidden

## Differentiation of Exponential Functions

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

*Example:* $\frac{d}{dx}(5^x) = 5^x \ln 5$

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

1. This is a product of two functions, so use the product rule $y' = u'v + uv'$, with $u=x^2$, $v=e^{3x}$
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 $x^n$.

*Calculator:* forbidden

## Differentiation of Logarithmic Functions

**Derivatives of logarithmic functions** — For natural logarithm ($x>0$): $\frac{d}{dx}(\ln x) = \frac{1}{x}$. For general logarithm ($a>0, a\neq 1, x>0$): $\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$.

*Example:* $\frac{d}{dx}(\log_{10} x) = \frac{1}{x \ln 10}$

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

1. Use the chain rule result for $\ln(g(x))$, where $g(x) = x^3 - 2x + 1$
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 $x^3 - 2x + 1 > 0$.

> **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.

*Calculator:* forbidden

## Common pitfalls

- **Wrong:** Writing $\frac{d}{dx}(\sin x) = -\cos x$ and $\frac{d}{dx}(\cos x) = \sin x$, swapping the signs.
  - Why it fails: The signs of the derivatives of sine and cosine are commonly mixed up during memorization.
  - Correct: Memorize the correct derivatives: $\frac{d}{dx}(\sin x) = \cos x$, $\frac{d}{dx}(\cos x) = -\sin x$.
- **Wrong:** Forgetting the chain rule for $e^{2x}$, writing $\frac{d}{dx}(e^{2x}) = e^{2x}$.
  - Why it fails: The inner function $2x$ has a derivative of 2 that must be included via the chain rule.
  - Correct: Apply the chain rule to get $\frac{d}{dx}(e^{2x}) = 2e^{2x}$.
- **Wrong:** Confusing derivatives of $e^x$ and $\ln x$, writing $\frac{d}{dx}(\ln x) = e^x$ or $\frac{d}{dx}(e^x) = \frac{1}{x}$.
  - Why it fails: These two core functions have distinct derivatives that are often swapped by students.
  - Correct: Remember: $\frac{d}{dx}(e^x) = e^x$ and $\frac{d}{dx}(\ln x) = \frac{1}{x}$.
- **Wrong:** Applying the power rule to $a^x$, writing $\frac{d}{dx}(a^x) = x a^{x-1}$.
  - Why it fails: The power rule applies to functions with variable base and constant exponent, not constant base and variable exponent.
  - Correct: Use the exponential derivative rule: $\frac{d}{dx}(a^x) = a^x \ln a$.

## Cheatsheet

| Function | Derivative |
| --- | --- |
| $\sin x$ | $\cos x$ |
| $\cos x$ | $-\sin x$ |
| $\tan x$ | $\sec^2 x$ |
| $e^x$ | $e^x$ |
| $a^x$ | $a^x \ln a$ |
| $\ln x$ | $\frac{1}{x}$ |
| $\log_a x$ | $\frac{1}{x \ln a}$ |
| $\ln(g(x))$ | $\frac{g'(x)}{g(x)}$ |
| $e^{g(x)}$ | $g'(x)e^{g(x)}$ |

## 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.

- [Stationary points, monotonicity and concavity](https://www.owlsprep.com/study/ib-math-aa-sl-u5-stationary-points-monotonicity-and-concavity/)
- [Optimization applications](https://www.owlsprep.com/study/ib-math-aa-sl-u5-optimization-applications/)
- [Indefinite integration and anti-differentiation](https://www.owlsprep.com/study/ib-math-aa-sl-u5-indefinite-integration-and-anti-differentiation/)

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