Differentiation Part1

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Contents

Notes

Learning Objectives (Syllabus)

Prior Knowledge

Notation

When we differentiate y=f(x)\, with respect to (w.r.t.) x\, we will obtain the derivative which can be written in any of the following form

Basic Formulas

Constants

A) \frac{d}{dx}c =0\, (c\, is a constant)

X power n

B) \frac{d}{dx}{x}^{n} =n{x}^{n-1}\,

Examples

Useful Formulas

These should be memorized

Plus/Minus/Multiply with Constant

C) \frac{d}{dx}\left[f\left(x\right)\pm g\left(x\right)\right] =\frac{d}{dx}\left[f\left(x\right)\right]\pm\frac{d}{dx}\left[g\left(x\right)\right]

D) \frac{d}{dx}c\left[f\left(x\right)\right] =c\frac{d}{dx}\left[f\left(x\right)\right](c\, is a constant)

Example

y=3{x}^{2}+\left(\pi+1\right)x-\frac{2}{x}+{\pi}^{2}-4{a}^{3}+1 where a\, is a constant.

Writing the Working

When writing the working out, always remember the function and its derivative are two DIFFERENT things. Thus, we must write clearly (and be clear about it ourselves) when we actually differentiate (as oppose to rearranging/simplifying) the given function.

e^x, ln x, sin x, cos x, tan x

Note : We don't really need to know the why's of each formula below, just need to memorize it.

E) \frac{d}{dx}{e}^{x} ={e}^{x}\,

F) \frac{d}{dx}{a}^{x} ={a}^{x}\ln a \,

G) \frac{d}{dx}\ln x =\frac{1}{x}

H) \frac{d}{dx}\sin x =\cos x \,

I) \frac{d}{dx}\cos x =-\sin x \,

J) \frac{d}{dx}\tan x ={\sec}^{2}x \,

Examples

Find f'\left(x\right) for the following

Exercise 1

Differentiate the following w.r.t. x\,

a) f\left(x\right)={x}^{7}+5{x}^{2}-3{x}^{-2}+3{x}^{\frac{1}{2}}

b) f\left(x\right)=2{x}^{5}-4x+\frac{3}{x}+5\sqrt{x}-\frac{2}{\sqrt{x}}

c) f\left(x\right)=2\ln x-{e}^{x}+{2}^{x}-{3}^{x}

d) f\left(x\right)=4\tan x+7\cos x-3\sin x

e) f\left(x\right)=\frac{2}{{x}^{4}}-\frac{\ln x}{4}+\frac{2\tan x}{3}

Products and Quotients

K) if y=uv\,, where u,v\, are functions of x\,

L) if y=\frac{u}{v}, where u,v\, are functions of x\,

Differentiating Complicated Functions

When u,v\, are complicated functions, we can differentiate it separately away from the main workings.

Say, for example, we have {e}^{\sin x}\cos \left(ax+b\right), we can see it's something times something {\color{Red}{e}^{\sin x}}{\color{Blue}\cos \left(ax+b\right)}. Differentiating those parts aren't really that difficult once we get the hang of it, but it's very easy to make careless mistakes if we do it too quickly. My suggestion is to do some side working.

Examples

Differentiate the following w.r.t x\,

Exercise 2

Differentiate the following w.r.t. x\,

a) y=\left(3x+1\right)\ln x

b) y={x}^{2}\cos x\,

c) y ={e}^{x}\sin x\,

d) y =x{e}^{x}\cos x \,

e) y =\left({e}^{x}-3\right)\left(\sin x+2x \right)

f) y ={e}^{x}\left(2\cos x-3{x}^{2} \right)

g) y =\frac{x+1}{x+2}

h) y =\frac{6{x}^{3}}{{x}^{2}-1}

i) y =\frac{1-x}{\sqrt{x+1}}

j) y =\frac{\ln x}{x}

k) y=\frac{{e}^{x}}{{x}^{3}}

l) y=\frac{x+1}{\sin x}

m) y=\frac{\sin x}{\sqrt{x}}

n) y=\frac{x}{2\ln x}

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