Differentiating Powers of x (Cambridge (CIE) A Level Maths: Pure 1): Revision Note

Differentiating powers of x

How do I differentiate expressions with powers of x?

• Powers of x are differentiated according to the following formula

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

    • Where $n$ is any real constant

  • This can be written as:

    • If $\mathrm{f}\left(x\right)=x^n$ then $\mathrm{f}\text{'}\left(x\right)=n{x}^{n-1}$

    • If $y=x^n$ then $\frac{dy}{dx}=n{x}^{n-1}$

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

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

    • 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

    • $\frac{d}{dx}\left(ax\right)=a$

    • Where $a$ is any real constant

  • The derivative of a constant function is zero

    • $\frac{d}{dx}\left(a\right)=0$

    • 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