Question

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

Answer 1

$Answer: \frac{\partial y}{\partial x}=8x^{8sinx}(cosx.logx+\frac{sinx}{x})$

Step-by-step 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,

$\frac{1}{y}.\frac{\partial y}{\partial x}=8(\frac{\partial sinx}{\partial x}.logx+sinx \frac{\partial logx}{\partial x})$

$\\\frac{1}{y}\frac{\partial y}{\partial x}=8(cosx.logx+sinx.\frac{1}{x})$

$\\\frac{\partial y}{\partial y}=8y(cosx.logx+\frac{sinx}{x})\\\\\frac{\partial y}{\partial x}=8x^{8sinx}(cosx.logx+\frac{sinx}{x})$