Differentiating Exponential & Logarithmic Functions (DP IB Analysis & Approaches (AA)): Revision Note

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Paul

Last updated

5 June 2025

Differentiating Exponential & Logarithmic Functions

What are exponential and logarithmic functions?

Exponential functions have term(s) where the variable ( $x$ ) is the power (exponent)
- In general, these would be of the form $y = a^{x}$
  - The special case of this is when $a = e$ , i.e. $y = e^{x}$
Logarithmic functions have term(s) where the logarithms of the variable ( $x$ ) are involved
- In general, these would be of the form $y = \log_{a} x$
  - The special case of this is when $a = e$ , i.e. $y = \log_{e} x = \ln x$

What are the derivatives of exponential functions?

The first two results, of the special cases above, have been met before
- $f (x) = e^{x}, f' (x) = e^{x}$
- $f (x) = \ln x, f' (x) = \frac{1}{x}$
- These are given in the formula booklet
For the general forms of exponentials and logarithms
- $f (x) = a^{x}$
  - $f' (x) = a^{x} (\ln a)$
- $f (x) = \log_{a} x$
  - $f' (x) = \frac{1}{x \ln a}$
- These are also given in the formula booklet

How do I show or prove the derivatives of exponential and logarithmic functions?

For $y = a^{x}$
- Take natural logarithms of both sides, $\ln y = x \ln a$
- Use the laws of logarithms, $\ln y = x \ln a$
- Differentiate, implicitly, $\frac{1}{y} \frac{d y}{d x} = \ln a$
- Rearrange, $\frac{d y}{d x} = y \ln a$
- Substitute for $y$ , $\frac{d y}{d x} = a^{x} \ln a$
For $y = \log_{a} x$
- Rewrite, $x = a^{y}$
- Differentiate $x$ with respect to $y$ , using the above result, $\frac{d x}{d y} = a^{y} \ln a$
- Using $\frac{d y}{d x} = \frac{1}{\frac{d x}{d y}}$ , $\frac{d y}{d x} = \frac{1}{a^{y} \ln a}$
- Substitute for $y$ , $\frac{d y}{d x} = \frac{1}{a^{\log_{a} x} \ln a}$
- Simplify, $\frac{d y}{d x} = \frac{1}{x \ln a}$

What do the derivatives of exponentials and logarithms look like with a linear functions of x?

For linear functions of the form $p x + q$
- $f (x) = a^{p x + q}$
  - $f' (x) = p a^{p x + q} (\ln a)$
- $f (x) = \log_{a} (p x + q)$
  - $f' (x) = \frac{p}{(p x + q) \ln a}$
- These are not in the formula booklet
  - they can be derived from chain rule
  - they are not essential to remember

Examiner Tips and Tricks

For questions that require the derivative in a particular format, you may need to use the laws of logarithms
- With ln appearing in denominators be careful with the division law
  - $\ln (\frac{a}{b}) = \ln a - \ln b$
  - but $\frac{\ln a}{\ln b}$ cannot be simplified (unless there is some numerical connection between $a$ and $b$ )

Worked Example

a) Find the derivative of $a^{3 x - 2}$ .

Chain rule or ' $p x + q$ shortcut' is required

$\frac{d}{d x} [a^{3 x - 2}] = a^{3 x - 2} \ln a \times 3$

$∴$ The derivative of $a^{3 x - 2}$ is $3 a^{3 x - 2} \ln a$

b) Find an expression for $\frac{d y}{d x}$ given that $y = \log_{5} (2 x^{3})$

Chain rule is needed

$\frac{d y}{d x} = \frac{1}{2 x^{3} \ln 5} \times 6 x^{2}$

Simplify by cancelling

$∴ \frac{d y}{d x} = \frac{3}{x \ln 5}$

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Previous:Differentiating Inverse Trigonometric FunctionsNext:Integrating with Reciprocal Trigonometric Functions

Differentiating Exponential & Logarithmic Functions (DP IB Analysis & Approaches (AA)): Revision Note

Differentiating Exponential & Logarithmic Functions

What are exponential and logarithmic functions?

What are the derivatives of exponential functions?

How do I show or prove the derivatives of exponential and logarithmic functions?

What do the derivatives of exponentials and logarithms look like with a linear functions of x?

You've read 0 of your 5 free revision notes this week

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

Number & Algebra Toolkit

Standard Form

Laws of Indices

Partial Fractions

Exponentials & Logs

Introduction to Logarithms

Laws of Logarithms

Solving Exponential Equations

Sequences & Series

Language of Sequences & Series

Sigma Notation

Arithmetic Sequences & Series

Geometric Sequences & Series

Applications of Sequences & Series

Compound Interest & Depreciation

Simple Proof & Reasoning

Proof by Deduction

Disproof by Counter Example

Proof by Induction & Contradiction

Proof by Induction

Proof by Contradiction

Binomial Theorem

Binomial Coefficients & Pascal's Triangle

Binomial Theorem

Extension of The Binomial Theorem

Permutations & Combinations

Counting Principles

Permutations

Combinations

Complex Numbers

Introduction to Complex Numbers

Operations with Complex Numbers

Modulus & Argument

Introduction to Argand Diagrams

Further Complex Numbers

Geometry of Complex Numbers

Modulus-Argument (Polar) Form

Exponential (Euler's) Form

Conversion between Forms of Complex Numbers

Complex Roots of Polynomials

De Moivre's Theorem

Proof of De Moivre's Theorem

Roots of Complex Numbers

Systems of Linear Equations

Systems of Linear Equations

Solving Systems Using Row Reduction

Number of Solutions to a System

2. Functions

Linear Functions & Graphs

Equations of a Straight Line

Parallel & Perpendicular Lines

Quadratic Functions & Graphs

Quadratic Functions

Factorising Quadratics

Completing the Square

Solving Quadratic Equations

Quadratic Inequalities

Discriminants

Functions Toolkit

Language of Functions

Composite Functions

Inverse Functions

Odd & Even Functions

Periodic Functions

Self-Inverse Functions

Graphing Functions & Their Key Features

Intersecting Graphs

Other Functions & Graphs

Exponential & Logarithmic Functions

Solving Equations Analytically