Question

What is the derivative of `y'` given `y = xsqrt(x) + 1/(x^2 sqrt(x))` ?

Answer 1

`y'' = 1/4(3x^(-1/2) +35 x^(-9/2))`

Explanation:

`y = xsqrt(x) + 1/(x^2 sqrt(x))`

We are asked to find the derivative of `y'` which is the second derivative of `y`, represented by `(d^2y)/dx^2` or `y''`

First rewrite the expression for `y` in term of powers of x `->`

`y = x^(3/2) + x^(-5/2)`

Using the power rule `->`

`y' = 3/2x^(1/2) - 5/2 x^(-7/2)`

Using the power rule again `->`

`y'' = 3/2 * 1/2 x^(-1/2) - 5/2 * (-7/2) x^(-9/2)`

`y'' = 3/4 x^(-1/2) + 35/4 x^(-9/2)`

`y'' = 1/4(3x^(-1/2) +35 x^(-9/2))`