topic badge
iGCSE (2021 Edition)

26.03 Power rule (ax^n) (Extended)

Lesson

We have seen how to find the derivative of functions of the form $f(x)=x^n$f(x)=xn using the power rule but what about multiples or combinations of these functions such as: $f(x)=5x^2$f(x)=5x2, and  $g(x)=3x^3+5x^2-6$g(x)=3x3+5x26

 

Rules for derivatives

Power Rule:

For a function $f(x)=x^n$f(x)=xn, $f'(x)=nx^{n-1}$f(x)=nxn1, for $n$n any real number.

 

Derivative of a constant multiple:

Derivative of constant multiple of a function is the multiple of the derivative. That is if $f(x)=kg(x)$f(x)=kg(x), where $k$k is a constant, then $f'(x)=kg'(x)$f(x)=kg(x)

In particular, if $f(x)=ax^n$f(x)=axn, then $f'(x)=nax^{n-1}$f(x)=naxn1, where $a$a is a constant.

 

Derivative of a sum or difference:

Derivative of sum is equal to the sum of the derivatives. That is if $f(x)=g(x)\pm h(x)$f(x)=g(x)±h(x) then $f'(x)=g'(x)\pm h'(x)$f(x)=g(x)±h(x)

This means we can differentiate each individual term. 

 

Worked examples

Example 1

Find the derivatives of the following functions.

Think: These are all multiples of power functions, hence use $f'(x)=anx^{n-1}$f(x)=anxn1.

(a) $f(x)=3x^2$f(x)=3x2

$f'(x)$f(x) $=$= $3\times2x$3×2x
  $=$= $6x$6x

(b) $g(m)=2m^4$g(m)=2m4

$g'(m)$g(m) $=$= $2\times4m^3$2×4m3
  $=$= $8m^3$8m3

Example 2

Find the derivative of the following functions.

Think: These functions can be considered as the sum of the function for each individual term. So we can differentiate the whole by finding the derivative of each part using the power and constant multiple rules.

(a) $f(x)=4x^2+3x+2$f(x)=4x2+3x+2

Thus, $f'(x)=8x+3$f(x)=8x+3   (remember that the derivative of a constant term is $0$0)

(b) $f(x)=3x^3-3x^2$f(x)=3x33x2

Thus, $f'(x)=9x^2-6x$f(x)=9x26x

Our rules encountered so far do not tell us how to differentiate in the case where a function is the product of two functions or the quotient of two functions.

Careful!

$\frac{d}{dx}\left[f(x)\times g(x)\right]\ne f'(x)\times g'(x)$ddx[f(x)×g(x)]f(x)×g(x) There are special rules for products and quotients. For now when you come across a product of two functions expand to produce a function with all terms of the form $ax^n$axn. For a quotient perform the division and simplify using index laws. Remember you can split a fraction into individual terms with the same denominator.

 

Worked examples

Example 3

Find the derivative of the following functions.

(a) $f(x)=(x+3)(x-1)$f(x)=(x+3)(x1)

First expand:

$f(x)$f(x) $=$= $x^2+3x-x-3$x2+3xx3
  $=$= $x^2+2x-3$x2+2x3

 Then differentiate: $f'(x)=2x+2$f(x)=2x+2

(b) $g(x)=x(x+1)^2$g(x)=x(x+1)2

First expand: 

$g(x)$g(x) $=$= $x(x^2+2x+1)$x(x2+2x+1)
  $=$= $x^3+2x^2+x$x3+2x2+x

Then differentiate: $g'(x)=3x^2+4x+1$g(x)=3x2+4x+1

 

Practice questions

question 1

Differentiate $y=2x^3-3x^2-4x+13$y=2x33x24x+13.

question 2

Consider the function $y=\left(x+4\right)^2$y=(x+4)2

  1. Express the function $y$y in expanded form.

  2. Hence find the derivative $\frac{dy}{dx}$dydx of the function $y=\left(x+4\right)^2$y=(x+4)2

question 3

Consider the function $f\left(x\right)=x^3-4x$f(x)=x34x.

  1. Determine $f'\left(x\right)$f(x).

  2. Determine $f'\left(4\right)$f(4).

  3. Determine $f'\left(-4\right)$f(4).

  4. Determine the $x$x-coordinates of the points at which $f'\left(x\right)=71$f(x)=71.

    State your solutions on the same line, separated by a comma.

Outcomes

0580E2.13A

Understand the idea of a derived function. Use the derivatives of functions of the form ax^n, and simple sums of not more than three of these.

What is Mathspace

About Mathspace