Question

What is dy/dx if y = (x2 + 2)3(x3 + 3)2? A. 3(x2 + 2)2(x3 + 3)2 + 2(x2 + 2)2(x3 + 3) B. 2x(x3 + 3)2 + 3x2(x2 + 2)3 C. 6x(x2 + 2)2(x3 +3)2 + 6x2(x2 + 2)3(x3 + 3) D. 6(x2 + 2)2(x3 + 3)

Answer 1

We want to find $\displaystyle{ \frac{dy}{dx}$, for

            $\displaystyle{ y=(x^2+2)^3(x^3+3)^2$.

Recall the product rule: for 2 differentiable functions f and g, the derivative of their product is as follows:

                                  $(fg)'=f'g+g'f$.

Thus,

 $y'=[(x^2+2)^3]'[(x^3+3)^2]+[(x^3+3)^2]'[(x^2+2)^3]\\\\ =3(x^2+2)^2(x^3+3)^2+2(x^3+3)(x^2+2)^3$

Answer: A)      $3(x^2+2)^2(x^3+3)^2+2(x^3+3)(x^2+2)^3$.