Derivative for log I have the following problem: \(\displaystyle{\log{{\left({\frac{{{x}+{3}}}{{{4}-{x}}}}\right)}}}\) I

Nancy Richmond

Nancy Richmond

Answered question


Derivative for log
I have the following problem:
I need to graph the following function so I will need a starting point, roots, zeros, stationary points, inflection points and local minimum and maximum and I need to know where the function grows and declines.
I calculated roots zeros x+3=0,x=3 and roots 4x=0,x=4. Now I sort of know how to graph the function from here but how do I get the stationary points do I have to find the derivative of log(x+34x) or just (x+34x)
I don't fully understand how to find the derivative of log. Can i use the log(x)=1x rule here to get 1x+34x and then find stationary points here ?

Answer & Explanation

Korbin Rivera

Korbin Rivera

Beginner2022-03-27Added 11 answers

For the stationary points you need to find ddx(log(x+34x)). You need to use the chain rule here ddxlog(f(x))=1f(x)f(x) which would give:
Hence, the function has no stationary points. Also, log(x+34x) has zeros when x+34x=1 because log(1)=0. So for the zeroes you need to solve x+3=4x giving x=12. It will also have asymptotes where the derivative goes to infinity i.e. at x=3 and x=4, the first going to and the second to +
Laylah Hebert

Laylah Hebert

Beginner2022-03-28Added 15 answers

It may help to write
Since, by the chain rule, we have

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