11.03 Differentiating x^n for non-integer n
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:
For a function$f(x)=x^n$f(x)=xn, the derivative is:
$f'(x)=nx^{n-1}$f′(x)=nxn−1
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′=nxn−1 to differentiate this function, subtracting one from the fractional indicie.
Do:
| $y'$y′ | $=$= | $\frac{1}{3}x^{\frac{1}{3}-1}$13x13−1 |
| $y'$y′ | $=$= | $\frac{1}{3}x^{\frac{-2}{3}}$13x−23 |
| $y'$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}}$13√x3 |
Example 2
Differentiate $y=x^{2.3}$y=x2.3:
Think: We use the rule $y'=nx^{n-1}$y′=nxn−1 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)=6√x at the point $\left(25,\frac{6}{5}\right)$(25,65).
Denote this gradient by $f'\left(25\right)$f′(25).