Differentiating Other Functions (Cambridge (CIE) A Level Maths: Pure 3): Revision Note

Differentiating other functions

How do I differentiate common functions?

• These are the common results

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

  • $\frac{d}{dx}\left({\mathrm{e}}^x\right)={\mathrm{e}}^x$

  • $\frac{d}{dx}\left(a^x\right)=a^x\ln a$ for $a>0$

  • $\frac{d}{dx}\left(\ln x\right)=\frac{1}{x}$

  • $\frac{d}{dx}\left(\sin x\right)=\cos x$

  • $\frac{d}{dx}\left(\cos x\right)=-\sin x$

  • $\frac{d}{dx}\left(\tan x\right)={\sec }^{2}x$

  • $\frac{d}{dx}\left(\cot x\right)=-\cos {\mathrm{ec}}^{2}x$

  • $\frac{d}{dx}\left(\sec x\right)=\sec x\tan x$

  • $\frac{d}{dx}\left(\cos \mathrm{ec}x\right)=-\cos \mathrm{ec}x\cot x$

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:

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

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

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

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