Use the product rule to find f(x) and then evaluate at the following points:

Question

Let f(x) = (4x³ +7x + 5)(4x^-2 - 6x^-3. Use the product rule to find f(x) and then evaluate at the following points:

(A) f′(12) = 

(B) f′(−12) = 

Answer 1

Let f(x) = (4x^3 +7x + 5)(4x^2 - 6x^3). Use the product rule to find f(x) and then evaluate at the following points: (A) f^' (12)= (B) f^' (-12)=

Let f(x) = (4x^3 + 7x + 5)(4x^2 - 6x^3). To find the derivative, use the product rule:

\[ f'(x) = (4x^3 + 7x + 5)'(4x^2 - 6x^3) + (4x^3 + 7x + 5)(4x^2 - 6x^3)'. \]

Now, differentiate each term:

\[ f'(x) = (12x^2 + 7)(4x^2 - 6x^3) + (4x^3 + 7x + 5)(8x - 18x^2). \]

Simplify the expression:

\[ f'(x) = 48x^4 - 72x^5 + 48x^2 - 72x^3 + 28x + 40x^3 - 54x^4 - 90x^2. \]

Combine like terms:

\[ f'(x) = -54x^5 - 26x^4 - 14x^3 - 42x^2 + 28x. \]

Now, evaluate at the given points:

(A) \( f'(12) = -54(12)^5 - 26(12)^4 - 14(12)^3 - 42(12)^2 + 28(12) \).

(B) \( f'(-12) = -54(-12)^5 - 26(-12)^4 - 14(-12)^3 - 42(-12)^2 + 28(-12) \).

You can now calculate these values for the final results.