Differentiating Reciprocal Trigonometric Functions (DP IB Analysis & Approaches (AA): HL): Revision Note

Differentiating reciprocal trigonometric functions

What are the reciprocal trigonometric functions?

• Secant, cosecant and cotangent are abbreviated and defined as

$\sec x=\frac{1}{\cos x}$           $\mathrm{cosec}x=\frac{1}{\sin x}$           $\cot x=\frac{1}{\tan x}$

• Remember that for calculus, angles need to be measured in radians

  • $\theta$ may be used instead of $x$

• $\mathrm{cosec}x$ is sometimes further abbreviated to $\csc x$

What are the derivatives of the reciprocal trigonometric functions?

• $f\left(x\right)=\sec x$

  • $f^{\text{'}}\left(x\right)=\sec x\tan x$

• $f\left(x\right)=\mathrm{cosec}x$
  

  • $f^{\text{'}}\left(x\right)=-\mathrm{cosec}x\cot x$

• $f\left(x\right)=\cot x$

  • $f^{\text{'}}\left(x\right)=-{\mathrm{cosec}}^{2}x$

How do I show or prove the derivatives of the reciprocal trigonometric functions?

• For $y=\sec x$

  • Rewrite, $y=\frac{1}{\cos x}$

  • Use quotient rule, $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\cos x\left(0\right)-\left(1\right)(-\sin x)}{{\cos }^{2}x}$

  • Rearrange, $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\sin x}{{\cos }^{2}x}$

  • Separate, $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{1}{\cos x}\times \frac{\sin x}{\cos x}$

  • Rewrite, $\frac{\mathrm{d}y}{\mathrm{d}x}=\sec x\tan x$

• Similarly, for $y=\mathrm{cosec}x$

  • $y=\frac{1}{\sin x}$

  • $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{\sin x\left(0\right)-\left(1\right)\cos x}{{\sin }^{2}x}$

  • $\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{-\cos x}{{\sin }^{2}x}$

  • $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{\sin x}\times \frac{\cos x}{\sin x}$

  • $\frac{\mathrm{d}y}{\mathrm{d}x}=-\mathrm{cosec}x\cot x$

• For the derivative of $y=\cot x$, see the Worked Example

What do the derivatives of reciprocal trig functions look like with linear functions of x?

• For linear functions of the form ax+b

  • $f\left(x\right)=\sec \left(ax+b\right)$

    • $f^{\text{'}}\left(x\right)=a\sec \left(ax+b\right)\tan \left(ax+b\right)$

  • $f\left(x\right)=\mathrm{cosec}(ax+b)$
    

    • $f^{\text{'}}\left(x\right)=-a\mathrm{cosec}(ax+b)\cot \left(ax+b\right)$

  • $f\left(x\right)=\cot (ax+b)$

    • $f^{\text{'}}\left(x\right)=-a{\mathrm{cosec}}^2(ax+b)$

  • These are not given in the exam formula booklet

    • they can be derived from chain rule

    • they are not essential to remember