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Question

Mathematics Question on limits and derivatives

Find the derivative of 2x+2x−1x\sqrt{2}x+2\sqrt{x}-\frac{1}{x} ?

A

2+1/x(1−12x)\sqrt{2}+1/\sqrt{x}(1- \frac {1}{2} x)

B

2−1/x(1+12x)\sqrt{2}-1/\sqrt{x}(1+\frac {1}{2}x)

C

2−1/x(1−12x)\sqrt{2}-1/\sqrt{x}(1-\frac {1}{2} x)

D

2+1/x(1+12x)\sqrt{2}+1/\sqrt{x}(1+\frac {1}{2} x)

Answer

2+1/x(1+12x)\sqrt{2}+1/\sqrt{x}(1+\frac {1}{2} x)

Explanation

Solution

The derivative of 2x+2x−1x\sqrt{2}x+2\sqrt{x}-\frac{1}{\sqrt{x}}
=2+2⋅12x12−1+12x−12−1=\sqrt{2}+2\cdot \frac{1}{2}{{x}^{\frac{1}{2}-1}}+\frac{1}{2}{{x}^{-\frac{1}{2}-1}}
=2+x−12+12x−32=\sqrt{2}+{{x}^{\frac{-1}{2}}}+\frac{1}{2}{{x}^{\frac{-3}{2}}}
=2+1x+12xx=\sqrt{2}+\frac{1}{\sqrt{x}}+\frac{1}{2x\sqrt{x}}
=2+1x(1+12x)=\sqrt{2}+\frac{1}{\sqrt{x}}\left( 1+\frac{1}{2x} \right)