# Chain Rule, Leibnitz Notation and Inner and Outer Functions

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The Chain Rule states how to differentiate composite functions

If and be differentiable, then the derivative of the composite function is

In Leibnitz notation, if and , then .

For example,

or

The derivative of the exponential function is

3. Identify the inner function and the outer function in . Then find the derivative .

4. Identify the inner function and the outer function in the following: . Then find the derivative .

5. Identify the inner function and the outer function in the following: . Then find the derivative

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The Chain Rule, Liebnitz Notation and Inner and Outer Functions are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

###### Education

- BSc , Wuhan Univ. China
- MA, Shandong Univ.

###### Recent Feedback

- "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
- "excellent work"
- "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

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