To determine

To evaluate: The derivative dydx

Answer

The value of derivative of y is dydx=xsinx[cosxlnx+sinxx]     

Explanation

Given that:

1. y=xsinx

2. Use log to calculate dydx

Formula used:

1. Chain rule: dfdx=dfdududx

2. ln(ab)=blna

3. Product rule: uv'=u'v+uv'

Calculation:

We have to use log function to calculate dydx. So, first we use log to simplify the function y.

Consider,

y=xsinx

Taking log both sides, we get

lny=ln(xsinx)=sinxlnx                                                                                 (By using formula 2)

Now, differentiate it with respect to x, we get

ddx(lny)=ddx[sinxlnx]1ydydx= ddx[sinx]lnx+sinxddx[lnx]                                          (By using chain rule and product rule)=cosx×lnx+sinx×1x=cosxlnx+sinxx

This implies that

dydx=y[cosxlnx+sinxx]=xsinx[cosxlnx+sinxx]

Conclusion: Thus, the derivative of y is dydx=xsinx[cosxlnx+sinxx]