Question

Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = sin(x – 2) + x² 

Answer 1

y = sin(x – 2) + x² 
Differentiating w.r.t. x, we get
${\displaystyle \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}[\sin (x-2)+x^2]}$

= ${\displaystyle \frac{\text{d}}{\text{dx}}\left[\sin (x-2)\right]+\frac{\text{d}}{\text{dx}}\left(x^2\right)}$

= ${\displaystyle \cos (x-2).\frac{\text{d}}{\text{dx}}(x-2)+2x}$
= cos(x – 2).(1 – 0) + 2x
= cos(x – 2) + 2x
The derivative of inverse function of y = f(x) is given by
${\displaystyle \frac{\text{dx}}{\text{dy}}=\frac{1}{\left(\frac{\text{dy}}{\text{dx}}\right)}}$

= ${\displaystyle \frac{1}{\cos (x-2)+2x}}$

At x = ${\displaystyle 2,\frac{\text{dx}}{\text{dy}}}$

= ${\displaystyle {\left(\frac{1}{[\cos (x-2)+2x]}\right)}_{\text{at} x=2}}$

= ${\displaystyle \frac{1}{\cos 0+2\left(2\right)}}$

= ${\displaystyle \frac{1}{1+4}}$

= ${\displaystyle \frac{1}{5}}$.