Find the derivative.
f(x)=ln(8-e^-x)
I got -e^-x/8-e^-x
but it was wrong.
WHAT DID I DOOOO?!
You're close, but missing something.
You understand that the derivative of ln(x) is 1/x, so
you have (8-e^(-x)) in the denominator, which is the first step correctly done.
The next step is to apply the chain rule to derive the argument of log,namely
d(8-e^(-x))/dx
=(0-(-e^(-x)))
=+e^(-x))
So the final answer should be
+e^(-x))/(8-e^-x)
Another source of error is that you do not appear to use parentheses appropriately.
Even if you had given the answer as
e^-x/8-e^-x
it would be incorrect, because it means
e^-(x/8)-e^-x
properly written (especially for a computer answer), it should be:
(e^-x)/(8-e^-x)
or even
(e^(-x))/(8-e^(-x))
To find the derivative of the function f(x) = ln(8 - e^(-x)), you can use the chain rule. Let's go step by step:
Step 1: Identify the outer function and the inner function. In this case, the outer function is ln(x), and the inner function is 8 - e^(-x).
Step 2: Take the derivative of the outer function. The derivative of ln(x) with respect to x is 1/x.
Step 3: Use the chain rule. Multiply the derivative of the outer function by the derivative of the inner function.
For the inner function, let's differentiate it:
Derivative of 8 with respect to x is 0.
Derivative of e^(-x) with respect to x is -e^(-x).
Step 4: Apply the chain rule. Multiply the derivative of the outer function (1/x) by the derivative of the inner function (in this case, -e^(-x)).
Therefore, the derivative of f(x) = ln(8 - e^(-x)) is:
f'(x) = (1 / (8 - e^(-x))) * (-e^(-x))
So you were on the right track, but made a small error. The correct derivative is:
f'(x) = -e^(-x) / (8 - e^(-x))
I hope this clarifies things for you!