Question

Let f(x) = (6x² - 6)(x² - 2) / (x² + 6). Evaluate f′(4).
f′(4) = ?

Answer 1

To evaluate f'(4), we need to find the derivative of the given function using the quotient rule. Substitute x = 4 into the derivative expression to get the value of f'(4).

Step-by-step explanation:

To evaluate f'(4), we need to find the derivative of the given function f(x) = (6x^2 - 6)(x^2 - 2) / (x^2 + 6).

To do that, we can use the quotient rule for differentiation. According to the quotient rule, if we have a function in the form f(x) = g(x) / h(x), then f'(x) = (g'(x)h(x) - g(x)h'(x)) / (h(x))^2.

Applying the quotient rule to our function, we get:

f'(x) = [(12x(x^2-2) + (6x^2 - 6)(2x)] / (x^2 + 6)^2

Now, we can substitute x = 4 into f'(x) to evaluate the derivative at x = 4.

f'(4) = [(12(4)(4^2 - 2) + (6(4)^2 - 6)(2(4)] / (4^2 + 6)^2

(Hint: Simplify the numerator and denominator separately before performing the division.)