Differentiation Rules

Description

The next resource in our series on differentiation for high school students looks at differentiation rules. These notes contain explanations on differentiating x\n and differentiating functions. Examples of each type are presented in the form of solved equations.
Niamh Ryan
Note by Niamh Ryan, updated more than 1 year ago
Niamh Ryan
Created by Niamh Ryan over 7 years ago
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Page 1

Differentiating xn

This is the one of the most frequently used rules of differentiation.   If y=xn thendydx=nxn1

  This can be used to differentiate surds or fractions involving powers of x .   For example, x can also be written as x12 , allowing it to be easily differentiated using the above rule.   3x can also be written as 3x1 , so    ddx(3x)=3x2
 

Page 2

Differentiating functions

ddx is a linear operator.  This means it can be separated out over addition or subtraction and constants can be moved outside the operation. Therefore,  ddx(xm+xn)=ddx(xm)+ddx(xn)

  and ddx(kxm)=kddx(xm)
where k is any constant.

Page 3

Example 1

Example 1: Differentiate the following function: y=3x+4x2

Answer: As demonstrated on the previous page, we can treat the two terms in this function as if they were separate functions and their constants do not change with differentiation. First, rewrite the function using exponents instead of surds and fractions: y=3x12+4x2
and then differentiatie:dydx=3(12)x121+4(2)x21=32x128x3
This can be simplified to give: dydx=32x8x3

Page 4

Example 2

Example 2: Differentiate the following function: y=x23x4x+1

Answer: Where there's a fraction with x terms on top and bottom, factorise the top before continuing. This gives:y=(x4)(x+1)x+1
Clearly (x+1) can be factored from top and bottom to givey=x4
which is the same as y=x14x0
and then differentiate:dydx=1
 

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