slope of the function f(x) = x^4 - 3x^2

Thank you for that very comprehensive answer Alan. 

f(x) = x^4 - 3x^2

When you find the first derivative you are really finding the gradient of the tangent to the curve at any specific x value.

This is an extremely important concept for you to assimilate!

$$\\f(x)=x^4-3x^2\\\\
f'(x)=4x^3-6x$$

That is it.  f'(x) will give you the value of the gradient to the curve for any specific x value!