Question

Differentiate with respect to x from the first principle?

`f(x)`=`x^(2sinx)`

Answer 1

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

Explanation:

`y=f(x)`

`y=x^(2sin(x))`

The Power Rule for cannot be used here, because `x` is not raised to a constant power, rather it is raised to the power of another function. Whenever we encounter a function in the exponent, we should apply the natural logarithm to both sides :

`ln(y)=ln(x^(2sin(x)))`

Recall the exponent rule for logarithms, which states that `ln(x^a)=aln(x)`. This rule applies even if `a` is a function and not just a constant.

`ln(y)=2sin(x)ln(x)`

Differentiate both sides with respect to `x`. This means the chain rule will apply for differentiating `ln(y)`, giving us an instance of `dy/dx:`

`1/y * dy/dx=(2sin(x))/x+2ln(x)cos(x)`

Solve for `dy/dx` by multiplying both sides by `y`:

`dy/dx=y((2sin(x))/x+2cos(x)ln(x))`

Recall that `y=x^(2sin(x))`:

`dy/dx=x^(2sin(x))((2sin(x))/x+2cos(x)ln(x))`

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