Derivative, Product and Chain Rule

Question

If , find .

Find 

Answer 1

Hello Hildana, So the first step to solving a problem like this is too foil out the entire expression prior to differentiation.

So,

f(t)= (t^²+3t+3)(6t²+6)

f(t)=6x⁴+18x³+24x²+18x+18

The next step is to differentiate each term to find the derivative

1. f(t)=6x⁴+18x³+24x²+18x+18
2. f'(t)=d/dx( 6x⁴+18x³+24x²+18x+18)
3. f'(t)= d/dx6x⁴ + d/dx18x³ + d/dx24x² + d/dx18x + d/dx18
4. f'(t)= 24x³+54x²+48x+18

To find f'(2) we then substitute 2 in place of x and evaluate.

1. f'(2)= 24(2)³+54(2)²+48(2)+18
2. f'(2)=522

Additionally this problem can be solved with the product rules evaluating the left hand side of f(t), (t^²+3t+3), as f(x) and the left hand side of f(t) ,(6t²+6) as g(x).

Following the product rule thus

1. f'(t)= f'(x)(g(x))+f(x)(g'(x))
2. f'(t)= d/dx((t^²+3t+3)(6t²+6)+ (t^²+3t+3)d/dx(6t²+6)
3. f'(t)= (2t+3)(6t²+6) + (t^²+3t+3)(12t)
4. f'(t)= (12t³+18t²+12t+18)+(12t³+36t² +36t)
5. f'(t)= 24x³+54x²+48x+18

Solving for f'(2)

1. f'(2)= 24(2)³+54(2)²+48(2)+18
2. f'(2)=522