Calculus Derivatives

Question

Can someone help me with the following calculus question:

 

I need to find the derivative of y = (x^2 + 3)(x^3 + 4)

 

I have to do that in the following two ways:

(a) Using the Product Rule. (Note: Apply the rule but do not simplify.)
y' =

(b) By performing the multiplication first. (Note: Simplify completely after finding the derivative.)
y' =

Answer 1

You can break down the question into simple derivation of each parenthetical component and then apply the product rule.

v  -> dx/dt(x^2+3)=2x

u -> dx/dt(x^3+4)=3x

Product Rule -> v'u+vu'

Derivative using the product rule:
2x(x^3+4)+3x(x^2+3)

The second way to handle this is to simplify the equation by *foiling* before deriving.

Simplified: x^5+4x^2+3x^3+12

dx/dt(x^5+4x+3x^3+12)=5x^4+8x+9x^2