SOLUTION: Differentiate with respect to x X^2 over square root x^2-1

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Question 990668: Differentiate with respect to x
X^2 over square root x^2-1

Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

y%22%22=%22%22x%5E2%2Fsqrt%28x%5E2-1%29

y%22%22=%22%22x%5E2%2F%28x%5E2-1%29%5E%281%2F2%29

We could differentiate it as a quotient but it's a 
little easier to first turn it into a product:

y%22%22=%22%22x%5E2%28x%5E2-1%29%5E%28-1%2F2%29

%28dy%29%2F%28dx%29%22%22=%22%22x%5E2%28-1%2F2%29%28x%5E2-1%29%5E%28-3%2F2%29%282x%29%22%22%2B%22%22%28x%5E2-1%29%5E%28-1%2F2%29%2A%282x%29+

%28dy%29%2F%28dx%29%22%22=%22%22x%5E2%28-1%2Fcross%282%29%29%28x%5E2-1%29%5E%28-3%2F2%29%28cross%282%29x%29%22%22%2B%22%22%28x%5E2-1%29%5E%28-1%2F2%29%2A%282x%29+

%28dy%29%2F%28dx%29%22%22=%22%22-x%5E3%28x%5E2-1%29%5E%28-3%2F2%29%22%22%2B%22%22%28x%5E2-1%29%5E%28-1%2F2%29%2A%282x%29+

A clever trick we often play here is to factor out the smallest exponent, the
-3%2F2 power.

When we factor the -3%2F2 power out of the -1%2F2 power what's left inside is 
the difference of the exponents %28-1%2F2%29-%28-3%2F2%29=-1%2F2%2B3%2F2=2%2F2=1
It's just the first power!

We factor x%28x%5E2-1%29%5E%28-3%2F2%29 out:

%28dy%29%2F%28dx%29%22%22=%22%22x%28x%5E2-1%29%5E%28-3%2F2%29+%28-x%5E2%2B2%28x%5E2-1%29%5E1%29+

%28dy%29%2F%28dx%29%22%22=%22%22x%28x%5E2-1%29%5E%28-3%2F2%29+%28-x%5E2%2B2%28x%5E2-1%29%5E%22%22%29+

%28dy%29%2F%28dx%29%22%22=%22%22x%28x%5E2-1%29%5E%28-3%2F2%29+%28-x%5E2%2B2x%5E2-2%29+

%28dy%29%2F%28dx%29%22%22=%22%22x%28x%5E2-1%29%5E%28-3%2F2%29+%28x%5E2-2%29+

We write the negative exponent in the bottom as a positive exponent:

%28dy%29%2F%28dx%29%22%22=%22%22x%28x%5E2-2%29%2F%28x%5E2-1%29%5E%283%2F2%29+

Edwin