Question

What is the derivative of the following function? f(x)=ln(sqrt(x^sin(x)))

f(x)=ln(sqrt(x^sin(x)))

Answer 1

`f^(')(x) =(cos(x)ln(x)+sin(x)/x)/2`

Explanation:

The function given is:

`=>f(x) = ln(sqrt(x^(sin(x))))`

Let's first simplify `sqrt(x^sin(x))`.

Using the law: `u(x) = e^ln(u(x)) " and " ln(a^b) =bln(a)` we get:

`=>sqrt(x^sin(x))= sqrt(e^(sin(x)*ln(x)))`

Also: `sqrt(x) =x^(1/2)`, so:

`=>sqrt(e^(sin(x)*ln(x)))= color(blue)(e^((sin(x)*ln(x)) /2))`

Let's go back to our function now! We have now:

`=>f(x) =ln(color(blue)(e^((sin(x)*ln(x))/2))) =(sin(x)*ln(x))/2`

And now the product rule:

`=>f(x) =u*v <=> f'(x) =u'*v+u*v'`

Applying the product rule to our function:

`=>color(green)(f^(')(x) =(cos(x)ln(x)+sin(x)/x)/2)`

A slightly nicer equivalent form:

`=>f^(')(x) = (xcos(x)ln(x) +sin(x))/(2x)`