Help Math Homework

Question

Find f'(1) if f(x)=ln(square root x^3+8x)

Answer 1

Or if f(x)=ln√(x³+8x)=(1/2)ln(x³+8x)

then f'(x)=(1/2)[(3x²+8)/(x³+8x)]

Answer 2

Let me first rewrite the original function so that it's a little bit more readable:

f(x) = ln( x^3/2 + 8x )

Next, we can use the chain rule to find f'(x). This gives us:

f'(x) = ( (3/2)x^1/2 + 8 ) / ( x^3/2 + 8x )

Then we can plug in 1 for x to find f'(1):

f'(1) = ( (3/2) * 1^1/2 + 8 ) / ( 1^3/2 + 8 * 1) = 19/18