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

(a) Given function y = (x^2+3)(x^3+4)

Applly derivative to each side with respect of x.

dy/dx = d/dx[(x^2+3)(x^3+4)]

= (x^2+3)d/dx(x^3+4)+(x^3+4)d/dx(x^2+3)

= (x^2+3)3x^2+(x^3+4)2x

= 3x^4+9x^2+2x^4+8x

y' = 5x^4+9x^2+8x

(b) Given function y = (x^2+3)(x^3+4)

y = x^5+4x^2+3x^3+12

Applly derivative to each side with respect of x.

dy/dx = d/dx [x^5+4x^2+3x^3+12]

= d/dx(x^5)+d/dx(4x^2)+d/dx(3x^3)+d/dx(12)

= 5x^4+8x+9x^2+0

= 5x^4+9x^2+8x

y' = 5x^4+9x^2+8x