Question

Use logarithmic differentiation to find the derivative of the function.y = x^ 8 sin(x)

Answer 1

`Answer: (\partial y)/(\partial x)=8x^(8sinx)(cosx.logx+(sinx)/(x))`

Explanation:

Since we have given that

`y=x^(8sinx)`

By using logarithmic on both sides we get,

`log y= 8 sinx. logx`

(∵
`log(a^m)=m.loga`)

Now, differentiating on both sides ,we get,

`(1)/(y).(\partial y)/(\partial x)=8((\partial sinx)/(\partial x).logx+sinx (\partial logx)/(\partial x))`

`\\(1)/(y)(\partial y)/(\partial x)=8(cosx.logx+sinx.(1)/(x))`

`\\(\partial y)/(\partial y)=8y(cosx.logx+(sinx)/(x))\\\\(\partial y)/(\partial x)=8x^(8sinx)(cosx.logx+(sinx)/(x))`