Question

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

Answer 1

Use a little algebra and the power rule to find `d/dx(4x^3+x^2)/2=6x^2+x`.

Explanation:

Start by splitting this up into two fractions, like so:
`(4x^3+x^2)/2=(4x^3)/2+x^2/2=2x^3+x^2/2`

Now, onto finding the derivative. The sum rule says we can break `d/dx(2x^3+x^2/2)` into `d/dx(2x^3)+d/dx(x^2/2)`; in other words, we can take the derivative of a larger function piece by piece. We will evaluate both of these using the power rule:
`d/dx(x^n)=nx^(n-1)`

Beginning with `d/dx(2x^3)`:
`d/dx(2x^3)=3*2x^(3-1)=6x^2`

For `d/dx(x^2/2)`, we have:
`d/dx(x^2/2)=2*x^(2-1)/2=x^1=x`

Putting these results back together yields:
`d/dx(4x^3+x^2)/2=6x^2+x`