iGCSE (2021 Edition)

11.03 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}$13​x13​−1 $y'$y′ $=$= $\frac{1}{3}x^{\frac{-2}{3}}$13​x−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=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.

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

0606C14.2

Use the notations f'(x), f''(x), dy/dx, d^2y/dx^2 [=d/dx(dy/dx)].

0606C14.3A

Use the derivatives of the standard functions x^n (for any rational n), together with constant multiples, sums and composite functions.