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8.05 Differentiating x^n for non-integer n

Lesson

From the previous lesson, we learnt how to differentiate functions raised to the power $n$n, where $n$n is an integer. In this lesson, we will focus on how to differentiate functions raised to non-integer values. The same process and formula is applied, however, care is required when dealing with fractions and decimals.

To recap, the following formula is used to differentiate functions raised to a given power:

Rule to differentiate powers of x

For a function$f(x)=x^n$f(x)=xn, the derivative is:

 $f'(x)=nx^{n-1}$f(x)=nxn1 

Worked examples

Example 1

Differentiate $y=x^{\frac{1}{3}}$y=x13 leaving your answer in surd form.

Think: We use the rule: $y'=nx^{n-1}$y=nxn1 to differentiate this function, subtracting one from the fractional indicie.

Do:

$y'$y $=$= $\frac{1}{3}x^{\frac{1}{3}-1}$13x131
$y'$y $=$= $\frac{1}{3}x^{\frac{-2}{3}}$13x23
$y'$y $=$= $\frac{1}{3x^{\frac{2}{3}}}$13x23
$y'$y $=$= $\frac{1}{3x^{\frac{2}{3}}}$13x23
$y'$y $=$= $\frac{1}{3\sqrt{x^3}}$13x3
Example 2

Differentiate $y=x^{2.3}$y=x2.3:

Think: We use the rule $y'=nx^{n-1}$y=nxn1 to differentiate this function substracting one from the decimal indicie.

Do:

$y'=2.3x^{1.3}$y=2.3x1.3

 

Practice questions

Question 1

Determine the derivative of $y=x^{\frac{6}{5}}$y=x65.

Question 2

Find the derivative of $y=x^6+x^{\frac{1}{5}}$y=x6+x15.

Give your answer with positive indices.

Question 3

Find the gradient of $f\left(x\right)=\frac{6}{\sqrt{x}}$f(x)=6x at the point $\left(25,\frac{6}{5}\right)$(25,65).

Denote this gradient by $f'\left(25\right)$f(25).

 

 

Outcomes

MA11-1

uses algebraic and graphical techniques to solve, and where appropriate, compare alternative solutions to problems

MA11-5

interprets the meaning of the derivative, determines the derivative of functions and applies these to solve simple practical problems

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