# Finding Derivatives : Fundamental Theory of Calculus and Chain Rule

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Using the fundamental theorem of calculus and chain rule,

For example, letting the expression equal F(x), and G(u)

Determine the derivative:

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##### Solution Summary

The Fundamental Theory of Calculus and Chain Rule are applied to finding two derivatives. The solution is well presented.

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Solution. Let's consider all of the possible cases we can have in a 3 × 3 system.

1) Success. In order for a system to be successful through elimination, there may be two possibilities for the structure of the coefficient matrix.

a) Full Success. The matrix may contain three independent columns. This way, the columns span all of space, and the planes formed by the rows intersect at a single point. Example:

b) Temporary Failure. The matrix may need a row exchange to allow for a pivot ...

###### Education

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

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- "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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