Chapter 4.2, Problem 36E

To determine

To calculate: The derivative of y with respect to x .

Answer to Problem 36E

  $\frac{dy}{dx}=-4{\left(1-6x\right)}^{\left(\frac{-1}{3}\right)}$

Explanation of Solution

Given information: The equation of y in terms of x is given as,

  $y={\left(1-6x\right)}^{\left(\frac{2}{3}\right)}$

Formula Used: $\frac{d\left(x^n\right)}{dx}=n\times x^{\left(n-1\right)}$

Calculation: Using the derivation,

  $\begin{array}{l}\frac{dy}{dx}=\frac{d\left({\left(1-6x\right)}^{\frac{2}{3}}\right)}{dx}\\ \frac{dy}{dx}=\frac{2}{3}\times {\left(1-6x\right)}^{\left(\frac{2}{3}-1\right)}\times \frac{d\left(1-6x\right)}{dx}\\ \frac{dy}{dx}=\frac{2}{3}\times {\left(1-6x\right)}^{\left(\frac{-1}{3}\right)}\times \left(-6\right)\\ \frac{dy}{dx}=-4{\left(1-6x\right)}^{\left(\frac{-1}{3}\right)}\end{array}$

So the derivative of y with respect to x is $\frac{dy}{dx}=-4{\left(1-6x\right)}^{\left(\frac{-1}{3}\right)}$ .