- #1

Yankel

- 395

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I was trying to find derivatives of two functions containing square roots. I got answers which I believe should be correct, however, the answers in the book differ significantly. The first answer of mine was checked in MAPLE and found correct. My guess that the author made some algebraic manipulations but I can't seem to track it down and get to the same result. Can you kindly take a look ?

The functions are:

\[f(x)=(\sqrt{x}+\frac{1}{\sqrt{x}})^{10}\]

\[g(x)=\frac{\sqrt{x}}{\sqrt{x}-\sqrt{x-1}}\]

My answers:

\[f'(x)=5\cdot (\sqrt{x}+\frac{1}{\sqrt{x}})^{9}\cdot (\frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x^{3}}})\]

\[g'(x)=\frac{\frac{1}{2\sqrt{x}}\cdot (\sqrt{x}-\sqrt{x-1})-\sqrt{x}(\frac{1}{2\sqrt{x}}-\frac{1}{2\sqrt{x-1}})}{(\sqrt{x}-\sqrt{x-1})^{2}}\]Books answers:

\[f'(x)=\frac{5\cdot (x+1)^{9}\cdot (x-1)}{x^{6}}\]

\[g'(x)=1+\frac{2x-1}{2\sqrt{x^{2}-x}}\]Thank you !