Let
f(x)=((x^(6))*((x-8)^(2)))/((x2+4)^(7))
Use logarithmic differentiation to determine the derivative.
f '(x) =
f '(8) =
ln f = ln(x^6) + ln(x-8)^2 - ln(x^2+4)^7
= 6ln(x) + 2ln(x-8) - 7ln(x^2+4)
1/f f' = d/dx 6ln(x) + 2ln(x-8) - 7ln(x^2+4)
f ' = (6/x + 2/(x-8) - 14x/(x^2+4)) * (x^6 * (x-8)^2) / (x^2+4)^7
you can massage that in various ways. One way ends up as
-2(x-8)(x^5)(3x^3-32x^2-16x+96)/(x^2+4)^8
plug in x=8 to get the value. That (x-8) in the numerator makes it all 0.
Actually, if you hadn't been asked for f '(x), you could have saved all the work by noting that if (x-k)^n is a factor of f(x), and n > 1, then (x-k) will be a factor of each term of f '(x), so f '(k) will be zero.
To find the derivative of the function f(x), we can use logarithmic differentiation. Here's how we can do it step by step:
Step 1: Take the natural logarithm of both sides of the equation to simplify the function:
ln(f(x)) = ln(((x^6)*((x-8)^2))/((x^2+4)^7))
Step 2: Apply the logarithm rules to simplify the expression:
ln(f(x)) = ln(x^6) + ln((x-8)^2) - ln((x^2+4)^7)
Step 3: Use the properties of logarithms to further simplify the equation. For example, ln(a*b) = ln(a) + ln(b) and ln(a^n) = n*ln(a):
ln(f(x)) = 6ln(x) + 2ln(x-8) - 7ln(x^2+4)
Step 4: Differentiate both sides of the equation with respect to x:
(d/dx) ln(f(x)) = (d/dx) (6ln(x) + 2ln(x-8) - 7ln(x^2+4))
Step 5: Use the chain rule and differentiation rules to find the derivative:
(f'(x))/f(x) = (6(1/x) + 2(1/(x-8)) - 7(1/(x^2+4)))
Step 6: Multiply both sides of the equation by f(x) to isolate f'(x):
f'(x) = f(x) * (6(1/x) + 2(1/(x-8)) - 7(1/(x^2+4)))
Substitute the original function f(x) back in:
f'(x) = ((x^6)*((x-8)^2)/((x^2+4)^7)) * (6(1/x) + 2(1/(x-8)) - 7(1/(x^2+4)))
Now we have the derivative of f(x), f'(x), in terms of the original function.
To find f'(8), we substitute x = 8 into the expression we derived for f'(x):
f'(8) = ((8^6)*((8-8)^2)/((8^2+4)^7)) * (6(1/8) + 2(1/(8-8)) - 7(1/(8^2+4)))
Simplifying further, we get:
f'(8) = (0/12^7) * (6(1/8) + 2(1/0) - 7(1/68))
Since we have division by zero (1/0) in the equation, it is undefined. Therefore, f'(8) is undefined in this case.