Question

Find the derivative of the inverse of the following functions, and also find their value at the points indicated against them. y = 3x² + 2logx³ 

Answer 1

y = 3x² + 2logx³ 
= 3x² + 6logx
Differentiating w.r.t. x, we get
${\displaystyle \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}(3x^2+6\log x)}$

= ${\displaystyle 3\times 2x+6\times \frac{1}{x}}$

= ${\displaystyle 6x+\frac{6}{x}}$

= ${\displaystyle \frac{6x^2+6}{x}}$
The derivative of inverse function of y = f(x) is given by
${\displaystyle \frac{\text{dx}}{\text{dy}}=\frac{1}{\left(\frac{\text{dy}}{\text{dx}}\right)}}$

= ${\displaystyle \frac{1}{\left(\frac{6x^2+6}{x}\right)}}$

= ${\displaystyle \frac{x}{6x^2+6}}$

At x = ${\displaystyle 1,\frac{\text{dx}}{\text{dy}}}$

= ${\displaystyle {\left(\frac{x}{6x^2+6}\right)}_{at x=1}}$

= ${\displaystyle \frac{1}{6{\left(1\right)}^2+6}}$

= ${\displaystyle \frac{1}{12}}$.