Differentiating Trig Functions

Differentiating Trig Functions

A LevelAQAEdexcelOCRAQA 2022

Differentiating Trig Functions

We’re now going to look at how to differentiate the simple trig functions – that’s \textcolor{blue}{\sin}, \textcolor{limegreen}{\cos} and \textcolor{red}{\tan}.

Make sure you are happy with the following topics before continuing.

A Level AQA Edexcel OCR

Simple Trig Results

Below is a diagram showing the derivative and integral of the basic trigonometric functions, \sin x and \cos x.

A LevelAQAEdexcelOCR

General Trig Differentiation

Below is a table of three derivative results.

A LevelAQAEdexcelOCR

Using the Chain Rule with Trig Functions

We can also use the Chain Rule to differentiate more complex trig functions.

For example, say we have f(x) = \textcolor{blue}{\sin} (x^2 - x + 1).

Then we can set \textcolor{red}{u} = x^2 - x + 1 and f(x) = g(\textcolor{red}{u}) = \textcolor{blue}{\sin} \textcolor{red}{u}.

Then \dfrac{d\textcolor{red}{u}}{dx} = 2x - 1 and \dfrac{df(x)}{d\textcolor{red}{u}} = \textcolor{limegreen}{\cos} u.

So,

\dfrac{df(x)}{dx} = (2x - 1)\textcolor{limegreen}{\cos} (x^2 - x + 1)

A LevelAQAEdexcelOCR

From First Principles

We can also find the derivatives from first principles.

For example, let f(x) = \textcolor{limegreen}{\cos} x. Then

f'(x) = \lim\limits_{\textcolor{purple}{h} \to 0}\left( \dfrac{\textcolor{limegreen}{\cos} (x + \textcolor{purple}{h}) - \textcolor{limegreen}{\cos} x}{\textcolor{purple}{h}} \right)

= \lim\limits_{\textcolor{purple}{h} \to 0}\left( \dfrac{\textcolor{limegreen}{\cos} x \textcolor{limegreen}{\cos} \textcolor{purple}{h} - \textcolor{blue}{\sin} x \textcolor{blue}{\sin} h - \textcolor{limegreen}{\cos} x}{\textcolor{purple}{h}}\right)

= \lim\limits_{\textcolor{purple}{h} \to 0}\left( \dfrac{\textcolor{limegreen}{\cos} x(\textcolor{limegreen}{\cos} \textcolor{purple}{h} - 1) - \textcolor{blue}{\sin} x \textcolor{blue}{\sin} \textcolor{purple}{h}}{\textcolor{purple}{h}}\right)

Using small angle approximations, we have

f'(x) = \lim\limits_{\textcolor{purple}{h} \to 0}\left( \dfrac{-\dfrac{1}{2}\textcolor{purple}{h}^2 \textcolor{limegreen}{\cos} x - \textcolor{purple}{h}\textcolor{blue}{\sin} x}{\textcolor{purple}{h}}\right)

= \lim\limits_{\textcolor{purple}{h} \to 0}\left( \dfrac{-1}{2}\textcolor{purple}{h}\textcolor{limegreen}{\cos} x - \textcolor{blue}{\sin} x\right)

= -\textcolor{blue}{\sin} x

A LevelAQAEdexcelOCR

Example Questions

x = \tan y gives

\dfrac{dx}{dy} = \sec ^2 y

Then \dfrac{dy}{dx} = \cos ^2 y.

Let u = x^2 and y = \tan u.

Then

\dfrac{dy}{dx} = 2x \times \sec ^2 u = \dfrac{2x}{\cos ^2 x^2}

f'(x) = \lim\limits_{h \to 0}\left( \dfrac{\sin (kx + kh) - \sin kx}{h} \right)

 

= \lim\limits_{h \to 0}\left( \dfrac{\sin kx \cos kh + \sin kh \cos kx - \sin kx}{h}\right)

 

= \lim\limits_{h \to 0}\left( \dfrac{\sin kx(\cos kh - 1) + \sin kh \cos kx}{h}\right)

 

Using small angle approximations, we have

f'(x) = \lim\limits_{h \to 0}\left( \dfrac{-\dfrac{1}{2}(kh)^2 \sin kx + kh\cos kx}{h}\right)

 

= \lim\limits_{h \to 0}\left( \dfrac{-1}{2}kh\sin kx + k\cos kx\right)

 

= k\cos kx

Related Topics

MME

Inverse Trig Functions

A Level
MME

Reciprocal Trig Functions

A Level

Additional Resources

MME

Exam Tips Cheat Sheet

A Level
MME

Formula Booklet

A Level

Worksheet and Example Questions

Site Logo

Differentiation

A Level

You May Also Like...

A Level Maths Revision Cards

The best A level maths revision cards for AQA, Edexcel, OCR, MEI and WJEC. Maths Made Easy is here to help you prepare effectively for your A Level maths exams.

£14.99
View Product

A Level Maths – Cards & Paper Bundle

A level maths revision cards and exam papers for Edexcel. Includes 2022 predicted papers based on the advance information released in February 2022! MME is here to help you study from home with our revision cards and practise papers.

From: £22.99
View Product

Transition Maths Cards

The transition maths cards are a perfect way to cover the higher level topics from GCSE whilst being introduced to new A level maths topics to help you prepare for year 12. Your ideal guide to getting started with A level maths!

£8.99
View Product