Differentiating Other Functions (Cambridge International AS Mathematics)

Differentiating Other Functions (Trig, ln & e etc)

Differentiating other functions

How do I differentiate common functions?

• These are the common results

  • $\frac{d}{dx}(x^{n})=nx^{n-1}$
  • $\frac{d}{dx}(e^{x})=e^{x}$
  • $\frac{d}{dx}(a^{x})=a^{x}\mathrm{ln}a$ for $a>0$
  • $\frac{d}{dx}(\mathrm{ln}x)=\frac{1}{x}$
  • $\frac{d}{dx}(\mathrm{sin}x)=\mathrm{cos}x$
  • $\frac{d}{dx}(\mathrm{cos}x)=-\mathrm{sin}x$
  • $\frac{d}{dx}(\mathrm{tan}x)=\mathrm{sec}^{2}x$
  • $\frac{d}{dx}(\mathrm{cot}x)=-\mathrm{cos} \mathrm{ec}^{2}x$
  • $\frac{d}{dx}(\mathrm{sec}x)=\mathrm{sec}x\mathrm{tan}x$
  • $\frac{d}{dx}(\mathrm{cos} \mathrm{ec}x)=-\mathrm{cos} \mathrm{ec}x\mathrm{cot}x$

How do I differentiate e^kx, a^x, ln(kx) and a^kx?

• And for a^kx:

• This last formula can be derived from Formula 3 by using the chain rule

How do I differentiate reciprocal trigonometric functions?

• The formulae for the derivatives of the reciprocal trigonometric functions are:

$\frac{d}{dx}(\mathrm{sec}x)=\mathrm{sec}x\mathrm{tan}x$

$\frac{d}{dx}(\mathrm{cosec}x)=-\mathrm{cosec}x\mathrm{cot}x$

$\frac{d}{dx}(\mathrm{cot}x)=-\mathrm{cosec}^{2}x$

• You can derive the derivatives for sec, cosec, and cot using the chain rule and the derivatives of the basic trigonometric functions

Exam Hint All of the above are in the formula book, make sure you know how to find them. They will be particularly useful when working with identities and when integrating trigonometric functions.

Worked Example