Differentiating Special Functions (Cambridge (CIE) O Level Additional Maths): Revision Note

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Amber

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25 December 2024

Differentiating Trig Functions

How do I differentiate trig functions?

For calculus with trigonometric functions, angles must be measured in radians
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 $\frac{d y}{d x} = \sec^{2} x$
The following two sets of relationships can be derived using the chain rule, but are useful to know!
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 $\frac{d y}{d x} = f^{'} (x) \sec^{2} (f (x))$

Examiner Tips and Tricks

Remember that these rules only work in radians!

Worked Example

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

i. $f (x) = \sin x$

ii. $f (x) = \cos 2 x$

ii. $f (x) = 3 \sin 4 x - \cos (2 x - 3)$

$f' (x) = \cos x$

ii. Use the chain rule or remember that when $y = \cos (a x + b)$ , then $\frac{d y}{d x} = - a \sin (a x + b)$

$f' (x) = - 2 \sin (2 x)$

iii. Differentiate 'term by term'

$f' (x) = 3 (4 \cos 4 x) - (- 2 \sin (2 x - 3))$

$f' (x) = 12 \cos 4 x + 2 \sin (2 x - 3)$

b) $\begin{matrix} \end{matrix}$ Find the gradient of the tangent to the curve $y = \sin (2 x + \frac{π}{6})$ at the point where $x = \frac{π}{8}$ .

Gradient of tangent is equal to gradient of curve. To find the gradient, differentiate...

$\frac{d y}{d x} = 2 \cos (2 x + \frac{π}{6})$

... and substitute $x = \frac{π}{8}$ into the derivative

$\frac{d y}{d x} = 2 \cos (2 (\frac{π}{8}) + \frac{π}{6})$

Ensure that your calculator is in radians

$= \frac{\sqrt{6} - \sqrt{2}}{2} = 0.517638090 . . .$

Answer = $\frac{\sqrt{6} - \sqrt{2}}{2}$ , or 0.518 to 3 significant figures

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 $x > 0$
In addition, the results below can all be found using the chain rule but are worthwhile knowing, to save time in an exam
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 $\frac{d y}{d x} = \frac{f' (x)}{f (x)}$

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 $y = e^{- 3 x} + 2 \ln x$ .

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

Differentiate each term separately.

$\frac{d y}{d x} = - 3 e^{- 3 x} + 2 (\frac{1}{x})$

Substitute $x = 2$ into the derivative.

$\frac{d y}{d x} = - 3 e^{- 3 (2)} + 2 (\frac{1}{2})$

Rearrange to the required form.

$\begin{array}{rcl} \frac{d y}{d x} & = & - 3 e^{- 6} + 1 = 1 - 3 e^{- 6} \end{array}$

Do not use your calculator to evaluate this as the question asks for the answer given in this form.

$1 - 3 e^{- 6}$

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Previous:Introduction to DifferentiationNext:Chain Rule

Differentiating Special Functions (Cambridge (CIE) O Level Additional Maths): Revision Note

Differentiating Trig Functions

How do I differentiate trig functions?

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

Algebra & Functions

Functions

Language of Functions

Inverse Functions

Composite Functions

Quadratic Functions

Solving Quadratics by Factorising

Quadratic Formula

Completing the Square

Quadratic Equation Methods

Discriminants

Quadratic Graphs

Quadratic Inequalities

Factors of Polynomials

Operations with Polynomials

Polynomial Division

Factor & Remainder Theorem

Solving Cubic Equations

Equations, Inequalities & Graphs

Modulus Functions

Graphs of Cubic Polynomials

Simultaneous Equations

Linear Simultaneous Equations

Quadratic Simultaneous Equations

Logarithmic & Exponential Functions

Exponential Functions

Logarithmic Functions

Laws of Logarithms

Exponential Equations

Transforming Relationships to Linear Form

Coordinate Geometry

Straight-Line Graphs

Linear Graphs

Parallel & Perpendicular Lines

Coordinate Geometry of the Circle

Equation of a Circle

Tangents to Circles

Intersection of Two Circles

Trigonometry

Circular Measure

Radian Measure

Arcs & Sectors

Trigonometry

Trigonometric Functions

The Unit Circle

Graphs of Trigonometric Functions

Trigonometric Identities

Solving Trigonometric Equations

Trigonometric Proof

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Binomial Theorem

Binomial Theorem

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Language of Sequences & Series

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Vectors

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Introduction to Vectors

Vector Addition

Problem Solving using Vectors

Compose & Resolve Velocities

Calculus

Differentiation

Introduction to Differentiation

Differentiating Special Functions

Chain Rule

Product Rule