Differentiating Powers of x (Edexcel IAL Maths)

Spec point — 4.2

Differentiating powers of x

How do I differentiate expressions with powers of x?

• Powers of x are differentiated according to the following formula

  • $\frac{d}{d x} ( x^{n ) = n x^{n} - 1}$

    • Where $n$ is any real constant

  • This can be written as:

    • If $\mathrm{f} ( x ) = x^{n}$ then $\mathrm{f} ' ( x ) = n x^{n - 1}$
    • If $y=x^{n}$ then $\frac{dy}{dx}=nx^{n-1}$

Exam Hint $\mathrm{f} ' ( x )$ is usually used when a function, $\mathrm{f} ( x )$, has been defined. $\frac{dy}{dx}$ is used when a graph of a function is given in the form $y=...$ They both represent the derivative of an expression.

• If the term involves a scalar multiple, then you can differentiate the power of x and then multiply by the scalar

  • $\frac{d}{d x} ( a x^{n ) = a n x^{n} - 1}$

    • Where $n$ and $a$ are any real constants

• Differentiating terms with positive integer powers of x is straightforward

• But negative and fractional powers of x can also be differentiated this way

• Don't forget these two special cases:

  • The derivative of a linear function is a constant

    • $\frac{d}{d x} ( a x ) = a$
    • Where $a$ is any real constant

  • The derivative of a constant function is zero

    • $\frac{d}{d x} ( a ) = 0$
    • Where $a$ is any real constant

• These allow you to differentiate constants and linear terms in x

• If the expression involves a sum or difference of terms just differentiate one term at a time

Exam Hint

• Take extra care when differentiating negative and fractional powers of x as mishandling negative signs and fractions is a common way to lose marks in these sorts of questions.

Worked Example