How do I differentiate 4x^3 + 2x + x^4 with respect to x?

With differentiation and integration, we can take each term separately. In general, to difference a term involving x's with respect to x, you first multiply the power of x by the coefficient of x to find the new coefficient. You then decrease the power of x by one.

The first term in this question is 4x^3. The power of x is 3 and the coefficient of x is 4, so we multiply these together to get 12. This is the coefficient of the differential. We then decrease the original power, 3, by one. So 4x^3 differentiated is 12x^2. Similarly, 2x differentiates to become 2 (because x=x^1 and multiplying 2 and 1 gets 2, and decreasing one from this power gets 0, so we have 2x^0 and x^0=1), and x^4 differentiates to 4x^3. So the whole thing differentiated is 12x^2+2+4x^3.