26.02 Power rule (x^n) (Extended)

Lesson

Worksheet

Practice

Lesson

We now want to combine what we know about instantaneous rates of change, to be able to calculate the instantaneous rate of change at a point using the derivative - a function giving us the gradient of the tangent at any point on the graph. Differentiating a function helps us find the gradient function, which can then be used to find the gradient of the tangent at any point along the function (which is also the instantaneous rate of change at that point).

Functions of the type $f(x)=x^n$f(x)=xn are called power functions. Below are some rules to differentiate these types of functions:

The derivative of a constant is zero. (E.g. the derivative of $f(x)=5$f(x)=5 is $f'(x)=0$f′(x)=0)
The derivative of a linear function is a constant. (E.g. the derivative of $f(x)=3x+1$f(x)=3x+1 is $f'(x)=3$f′(x)=3)
The derivative of a quadratic function is a linear function. (E.g. the derivative of $f(x)=4x^2+3x+1$f(x)=4x2+3x+1 is $f'(x)=8x+3$f′(x)=8x+3)
The derivative of a cubic function is a quadratic function. (E.g. the derivative of $f(x)=x^3+6x$f(x)=x3+6x is $f'(x)=3x^2+6$f′(x)=3x2+6)
The derivative of a quartic function is a cubic function. (E.g. the derivative of $f(x)=2x^4$f(x)=2x4 is $f'(x)=8x^3$f′(x)=8x3)

This suggests that the derivative of a polynomial of degree $n$n is of degree $n-1$n−1, for integer values of $n\ge1$n≥1.

Looking further at just examples of functions of the form $f(x)=x^n$f(x)=xn, we have:

$f(x)$`f`(`x`)	$f'(x)$`f`′(`x`)
$x$`x`	$1$1
$x^2$`x`2	$2x$2`x`
$x^3$`x`3	$3x^2$3`x`2
$x^4$`x`4	$4x^3$4`x`3
$x^5$`x`5	$5x^4$5`x`4

Noticing the pattern in the table, we could make the conjecture that the derivative of $f(x)=x^n$f(x)=xn is $f'(x)=nx^{n-1}$f′(x)=nxn−1. This is in fact the case and we call this formula the power rule. The power rule applies to not only positive integer values of $n$n but for $n$n being any real number. Once we establish the rule we can use the power rule as a quick way to find the derivative.

Power rule

For a function $f(x)=x^n$f(x)=xn, the derivative is $f'(x)=nx^{n-1}$f′(x)=nxn−1, for $n$n any real number.

Worked examples

Example 1

Find the derivatives of the following functions using the power rule.

Think: The power rule tells us "bring the power to the front and then subtract one from the power".

(a) $f(x)=x^2$f(x)=x2

Thus, $f'(x)=2x$f′(x)=2x.

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

Thus, $g'(m)=4m^3$g′(m)=4m3.

Practice questions

Question 1

Find the gradient of $f\left(x\right)=x^4$f(x)=x4 at $x=2$x=2.

Denote this gradient by $f'\left(2\right)$f′(2).

Question 2

Use the applet below to explore how the gradient of the tangent changes at different points along $y=x^2$y=x2. Then answer the questions that follow.

Created with Geogebra

Which feature of the gradient function tells us whether $y=x^2$y=x2 is increasing or decreasing?
The gradient function is decreasing when $y=x^2$y=x2 is increasing, and increasing when $y=x^2$y=x2 is decreasing.
A
The gradient function is increasing when $y=x^2$y=x2 is increasing, and decreasing when $y=x^2$y=x2 is decreasing.
B
The gradient function is negative when $y=x^2$y=x2 is increasing, and positive when $y=x^2$y=x2 is decreasing.
C
The gradient function is positive when $y=x^2$y=x2 is increasing, and negative when $y=x^2$y=x2 is decreasing.
D
For $x>0$x>0, is the gradient of the tangent positive or negative?
Positive
A
Negative
B
For $x\ge0$x≥0, as the value of $x$x increases how does the gradient of the tangent line change?
The gradient of the tangent line increases at a constant rate.
A
The gradient of the tangent line increases at an increasing rate.
B
The gradient of the tangent line remains constant.
C
For $x<0$x<0, is the gradient of the tangent positive or negative?
Positive
A
Negative
B
For $x<0$x<0, as the value of $x$x increases how does the gradient of the tangent line change?
The gradient of the tangent line remains constant.
A
The gradient of the tangent line increases at a constant rate.
B
The gradient of the tangent line increases at an increasing rate.
C
Complete the following statement:

"For $y=x^2$y=x2, the gradient of the tangent line changes at a constant rate. This means the derivative $y'$y′ is a function."
cubic
A
linear
B
constant
C
quadratic
D

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.