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

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Paul

Last updated

11 December 2024

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Differentiating Trig Functions

How do I differentiate sin, cos and tan?

The derivative of $y = \sin x$ is $\frac{d y}{d x} = \cos x$
The derivative of $y = \cos x$ is $\frac{d y}{d x} = - \sin x$
The derivative of $y = \tan x$ is $\frac{d y}{d x} = \sec^{2} x$
- This result can be derived using quotient rule
For the linear function $a x + b$ , where $a$ and $b$ are constants,
- the derivative of $y = \sin (a x + b)$ is $\frac{d y}{d x} = a \cos (a x + b)$
- the derivative of $y = \cos (a x + b)$ is $\frac{d y}{d x} = - a \sin (a x + b)$
- the derivative of $y = \tan (a x + b)$ is $\frac{d y}{d x} = a \sec^{2} (a x + b)$
For the general function $f (x)$ ,
- the derivative of $y = \sin (f (x))$ is $\frac{d y}{d x} = f^{'} (x) \cos (f (x))$
- the derivative of $y = \cos (f (x))$ is $\frac{d y}{d x} = - f' (x) \sin (f (x))$
- the derivative of $y = \tan (f (x))$ is $\frac{d y}{d x} = f' (x) \sec^{2} (f (x))$
These last three results can be derived using the chain rule
For calculus with trigonometric functions angles must be measured in radians
- Ensure you know how to change the angle mode on your GDC

Examiner Tips and Tricks

As soon as you see a question involving differentiation and trigonometry put your GDC into radians mode

Worked Example

a) Find $f' (x)$ for the functions

$f (x) = \sin x$
$f (x) = \cos (5 x + 1)$

b) A curve has equation $y = \tan (6 x^{2} - \frac{π}{4})$ .

Find the gradient of the tangent to the curve at the point where $x = \frac{\sqrt{π}}{2}$ .

Give your answer as an exact value.

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Differentiating e^x & lnx

How do I differentiate exponentials and logarithms?

The derivative of $y = e^{x}$ is $\frac{d y}{d x} = e^{x}$ where $x \in ℝ$
The derivative of $y = \ln x$ is $\frac{d y}{d x} = \frac{1}{x}$ where $x > 0$
For the linear function $a x + b$ , where $a$ and $b$ are constants,
- the derivative of $y = e^{(a x + b)}$ is $\frac{d y}{d x} = a e^{(a x + b)}$
- the derivative of $y = \ln (a x + b)$ is $\frac{d y}{d x} = \frac{a}{(a x + b)}$
  - in the special case $b = 0$ , $\frac{d y}{d x} = \frac{1}{x}$ ( $a$ 's cancel)
For the general function $f (x)$ ,
- the derivative of $y = e^{f (x)}$ is $\frac{d y}{d x} = f' (x) e^{f (x)}$
- the derivative of $y = \ln (f (x))$ is $\frac{d y}{d x} = \frac{f' (x)}{f (x)}$
The last two sets of results can be derived using the chain rule

Examiner Tips and Tricks

Remember to avoid the common mistakes:
- the derivative of $\ln k x$ with respect to $x$ is $\frac{1}{x}$ , NOT $\frac{k}{x}$
- the derivative of $e^{k x}$ with respect to $x$ is $k e^{k x}$ , NOT $k x e^{k x - 1}$

Worked Example

A curve has the equation $y = e^{- 3 x + 1} + 2 \ln 5 x$ .

Find the gradient of the curve at the point where $x = 2$ giving your answer in the form $y = a + b e^{c}$ , where $a, b$ and $c$ are integers to be found.

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Previous:Applications of DifferentiationNext:Techniques of Differentiation

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

Differentiating Trig Functions

How do I differentiate sin, cos and tan?

Differentiating e^x & lnx

How do I differentiate exponentials and logarithms?

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

Sign up now. 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

Arithmetic Sequences & Series

Geometric Sequences & Series

Applications of Sequences & Series

Compound Interest & Depreciation

Simple Proof & Reasoning

Proof

Proof by Induction & Contradiction

Proof by Induction

Proof by Contradiction

Binomial Theorem

Binomial Theorem

Extension of The Binomial Theorem

Permutations & Combinations

Counting Principles

Permutations & Combinations

Complex Numbers

Introduction to Complex Numbers

Modulus & Argument

Introduction to Argand Diagrams

Further Complex Numbers

Geometry of Complex Numbers

Forms of Complex Numbers

Complex Roots of Polynomials

De Moivre's Theorem

Roots of Complex Numbers

Systems of Linear Equations

Systems of Linear Equations

Algebraic Solutions

2. Functions

Linear Functions & Graphs

Equations of a Straight Line

Quadratic Functions & Graphs

Quadratic Functions

Factorising & Completing the Square

Solving Quadratics

Quadratic Inequalities

Discriminants

Functions Toolkit

Language of Functions

Composite & Inverse Functions

Symmetry of Functions

Graphing Functions

Other Functions & Graphs

Exponential & Logarithmic Functions

Solving Equations

Modelling with Functions

Reciprocal & Rational Functions

Reciprocal & Rational Functions

Transformations of Graphs

Translations of Graphs

Reflections of Graphs

Stretches of Graphs

Composite Transformations of Graphs

Polynomial Functions

Factor & Remainder Theorem

Polynomial Division

Polynomial Functions

Roots of Polynomials

Inequalities

Solving Inequalities Graphically

Polynomial Inequalities