If f(x) = 2 \ln(8 x+4\ln(x)) find f'(2)
To find the derivative of the function f(x), we can use the chain rule of differentiation. The chain rule states that if we have a composite function,
g(f(x)), the derivative of g(f(x)) is equal to g'(f(x)) * f'(x).
In this case, our composite function is 2 ln(8x + 4 ln(x)). Let's break it down step by step:
Step 1: Rewrite the function using the properties of logarithms.
f(x) = 2 ln(8x + ln(x^4))
Step 2: Differentiate the function term by term.
f'(x) = 2 * d/dx(ln(8x + ln(x^4)))
Step 3: Apply the chain rule. Let u = 8x + ln(x^4).
So, f'(x) = 2 * dudx * d/dx(ln(u))
Step 4: Calculate dudx.
dudx = d(8x + ln(x^4))/dx
= 8 + d/dx(ln(x^4))
= 8 + 4 * 1/x
= 8 + 4/x
Step 5: Differentiate the natural logarithm term using d/dx(ln(u)) = 1/u * dudx.
d/dx(ln(u)) = 1/u * dudx
= 1/(8x + ln(x^4)) * (8 + 4/x)
Step 6: Substitute the values back into f'(x).
f'(x) = 2 * (8 + 4/x)/(8x + ln(x^4))
Step 7: Evaluate f'(2).
f'(2) = 2 * (8 + 4/2)/(8(2) + ln((2)^4))
= 2 * (8 + 2)/(16 + ln(16))
= 2 * 10/(16 + ln(16))
= 20/(16 + ln(16))
So, f'(2) = 20/(16 + ln(16)).