Question

How do you find the derivative of `4sqrt(x-3) + x^5/(2x)`?

Answer 1

The first step here would be to write the equation in a way that is easier to understand:

`f(x) =4sqrt(x-3) + x^5/(2x) = 4(x-3)^(1/2) + x^4/2`

now we solve for `f'(x)`

we can use the chain rule in the first section. The chain rule states that:

`(dy)/(du)*(du)/(dx)=(dy)/(dx)`

Naming `u=(x-3)` and then using the chain rule, we get:

`color(red)(d/(dx) 4(x-3)^(1/2) = 2(x-3)^(-1/2)(1) = 2(x-3)^(-1/2))`

and the second section would be:

`color(blue)(d/dx x^4/2 = 2x^3)`

now we also know that

`f'(x) = d/(dx) 4(x-3)^(1/2) + d/dx x^4/2`

so we can now just drop in our two results:

`f'(x) = 2(x-3)^(-1/2) + 2x^3`