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Expert: - 12/18/2009


Question
Hi Scotto, I was wondering if you could help me with 2 derivative problems?

1. derivative of f(x) = radical(3-5(ln)x)

2. derivative of y = (4-cosx)/(1+tanx)

Thank you so much, I really appreciate it!
-Carly

Answer
1. If f(x) = √(3-5•ln(x)), that is a chain rule.
The chain rule states that d(f(g(x))/dx = (df/dg)(dg/dx).

If the function were the √x, the derivative would be 1/(2√x).
From this we can say that the derivative of √(3-5•ln(x)) is 1/(2√(3-5•ln(x)))(d(3-5•ln(x))/dx).

The derivative of 3 - 5•ln(x) is -5/x.  Putting that in gives -5/(2x√(3-5•ln(x))).

2. If y = (4-cosx)/(1+tanx), the y' = (lo d hi - hi d lo)/lo˛.

The way to remember that is it has a girl in the second term - you know - hi d (as in Heidi).
You could also not that it starts and ends with lo, which resembles the bottom, lo˛.

It is known that hi = 4 - cosx, so d hi = cinx.
It is also known that lo = 1 + tanx, so d lo = sec˛x.

Putting hi, d hi, lo, and d lo in the equation given gives the answer.

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