Chapter 4.1, Problem 1E

To determine

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

Answer to Problem 1E

  $\frac{dy}{dx}=3\cos \left(3x+1\right)$ .

Explanation of Solution

Given information: The equation of y and u is given as,

  $\begin{array}{l}y=\sin \left(3x+1\right)\\ u=3x+1\end{array}$

Formula used: The formula that is used here is Chain Rule which is as follows,

  $\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}$ .............. (1)

First calculate the derivative of u with respect to x, then substitute it in the equation of y and then apply the Chain Rule.

Calculation: Substitute u in y ,

  $y=\sin u$

Apply the Chain Rule given in equation (1),

  $\begin{array}{l}\frac{dy}{dx}=\frac{dy}{du}\times \frac{du}{dx}\\ \frac{dy}{dx}=\frac{d\sin u}{du}\times \frac{d\left(3x+1\right)}{dx}\\ \frac{dy}{dx}=\cos u\times \left(3+0\right)\\ \frac{dy}{dx}=3\cos u\end{array}$

Substitute u,

  $\frac{dy}{dx}=3\cos \left(3x+1\right)$

The derivative of y with respect to x is $\frac{dy}{dx}=3\cos \left(3x+1\right)$ .