# What is the derivative of #x^sin(x)#?

##### 3 Answers

#### Explanation:

#y = x^sinx#

Take the natural logarithm of both sides.

#lny = ln(x^sinx)#

Use laws of logarithms to simplify.

#lny = sinxlnx#

Use the product rule and implicit differentiation to differentiate.

#1/y(dy/dx) = cosx(lnx) + 1/x(sinx)#

#1/y(dy/dx) = cosxlnx + sinx/x#

#dy/dx = (cosxlnx + sinx/x)/(1/y)#

#dy/dx = x^sinx(cosxlnx+sinx/x)#

Hopefully this helps!

#### Explanation:

When we have a function of **take the (natural) logarithm of both sides**:

#ln y = ln (x^(sin x))#

#color(white)(ln y)=sin x * ln x#

This places all the **take the derivative of both sides with respect to #x#**:

#=>d/dx (ln y)=d/dx (sin x * ln x)#

Remembering that

#=> 1/y*dy/dx=cos x * ln x + sin x (1/x)#

#=> color(white)"XXi"dy/dx=y[cos x * ln x + (sin x) /x]#

Since we began with

#=> color(white)"XXi"dy/dx=x^(sin x)[cos x * ln x + (sin x)/x]# .

## Note:

When

#### Explanation:

Given:

Use logarithmic differentiation.

On the right side, use a property of logarithms,

Use implicit differentiation on the left side:

Use the product rule on the right sides:

let

Substituting into the product rule:

Put the equation back together:

Multiply both sides by y:

Substitute