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]