Exercise
Find the derivative of the following function:
f(x)={(\sqrt{x}+\frac{1}{\sqrt{x}})}^{10}
Final Answer
Solution
We simplify the function before differentiating:
f(x)={(\sqrt{x}+\frac{1}{\sqrt{x}})}^{10}=
={(\frac{x+1}{\sqrt{x}})}^{10}
Using Derivative formulas and the quotient rule in Derivative Rules, we get the derivative:
f'(x)=10{(\frac{x+1}{\sqrt{x}})}^9\cdot \frac{\sqrt{x}-(x+1)\cdot\frac{1}{2\sqrt{x}}}{x}=
One can simplify the derivative:
=10\cdot\frac{{(x+1)}^9}{{(\sqrt{x})}^9}\cdot \frac{\sqrt{x}-\frac{x+1}{2\sqrt{x}}}{x}=
=10\cdot\frac{{(x+1)}^9}{{(\sqrt{x})}^9}\cdot\frac{2x-x-1}{2x\sqrt{x}}=
=\frac{5{(x+1)}^9(x-1)}{x^{\frac{9}{2}}\cdot x\cdot x^{\frac{1}{2}}}=
=\frac{5{(x+1)}^9(x-1)}{x^6}
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