Question

How do you find the derivative of `y=(1/3) ((x^2)+2)^(3/2)`?

Answer 1

`dy/dx=xsqrt(x^2+2)`

Explanation:

Rewriting slightly:

`y=1/3(x^2+2)^(3/2)`

Notice that `(x^2+2)^(3/2)` is a function in the form `u^(3/2)`. The power rule and the chain tells us that we differentiate this normally, in that the derivative of `x^(3/2)` is `3/2x^(1/2)`, but when there's a function in the middle we will multiply by the derivative of the inner function. Thus, the derivative of `u^(3/2)` is `3/2u^(1/2)*d/dx(u)`.

So, we see that:

`dy/dx=1/3(3/2(x^2+2)^(1/2))*d/dx(x^2+2)`

The power rule tells us that the derivative of `x^2+2` is `2x`:

`dy/dx=1/3(3/2)(2x)(x^2+2)^(1/2)`

Simplifying:

`dy/dx=xsqrt(x^2+2)`