Question

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

Answer 1

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

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!

Answer 2

`d/dx x^(sin x)=x^(sin x)[cos x * ln x + (sin x)/x]`.

Explanation:

When we have a function of `x` like `y=x^sin x`, where a single term contains `x` in both its base and its power, perhaps the easiest way to find the function's derivative is to first 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 `x`'s on the same "level". Then, take the derivative of both sides with respect to `x`:

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

Remembering that `y` is a function of `x`, we get

`=> 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 `y=x^(sin x)`, we substitute this back in for `y` to get

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

Note:

When `f(x)=g(x)^(h(x))`, you'll almost always see `g(x)^(h(x))` appear in the derivative of `f(x)`. If you don't, go back and double check your work to make sure things were done right.

Answer 3

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

Explanation:

Given: `y = x^(sin(x))`

Use logarithmic differentiation.

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

On the right side, use a property of logarithms, `ln(a^b) = (b)ln(a)`:

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

Use implicit differentiation on the left side:

`(dln(y))/dx = 1/ydy/dx`

Use the product rule on the right sides:

`(d(uv))/dx = (u')(v) + (u)(v')`

let `u = sin(x)`, then `u' = cos(x), v =ln(x), and v' = 1/x`

Substituting into the product rule:

`(d((sin(x))ln(x)))/dx = (cos(x))ln(x) + (sin(x))/x`

Put the equation back together:

`1/ydy/dx = (cos(x))ln(x) + (sin(x))/x`

Multiply both sides by y:

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

Substitute `x^(sin(x))` for y:

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