What is the derivative of y=(x+1/x)/(2sqrt(x)+1/x) ?

The derivative of y=(x+1/x)/(2sqrt(x)+1/x) is (sqrt(x)(x^2+2sqrt(x)-3))/((2xsqrt(x)+1)^2)

What is the first derivative of y=(x+1/x)/(2sqrt(x)+1/x) ?

The first derivative of y=(x+1/x)/(2sqrt(x)+1/x) is (sqrt(x)(x^2+2sqrt(x)-3))/((2xsqrt(x)+1)^2)

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derivative of

y = \frac{x + \frac{1}{x}}{2 x + \frac{1}{x}}

\frac{x ( x ^{2} + 2 x - 3 )}{( 2 x x + 1 ) ^{2}}

Solution steps

\frac{d}{d x} (\frac{x + \frac{1}{x}}{2 x + \frac{1}{x}})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( x + \frac{1}{x} ) ( 2 x + \frac{1}{x} ) - \frac{d}{d x} ( 2 x + \frac{1}{x} ) ( x + \frac{1}{x} )}{( 2 x + \frac{1}{x} ) ^{2}}

\frac{d}{d x} (x + \frac{1}{x}) = 1 - \frac{1}{x ^{2}}

\frac{d}{d x} (2 x + \frac{1}{x}) = \frac{x ^{\frac{3}{2}} - 1}{x ^{\frac{3}{2}} x}

= \frac{( 1 - \frac{1}{x ^{2}} ) ( 2 x + \frac{1}{x} ) - \frac{x ^{\frac{3}{2}} - 1}{x ^{\frac{3}{2}} x} ( x + \frac{1}{x} )}{( 2 x + \frac{1}{x} ) ^{2}}

Simplify

\frac{( 1 - \frac{1}{x ^{2}} ) ( 2 x + \frac{1}{x} ) - \frac{x ^{\frac{3}{2}} - 1}{x ^{\frac{3}{2}} x} ( x + \frac{1}{x} )}{( 2 x + \frac{1}{x} ) ^{2}} : \frac{x ( x ^{2} + 2 x - 3 )}{( 2 x x + 1 ) ^{2}}

= \frac{x ( x ^{2} + 2 x - 3 )}{( 2 x x + 1 ) ^{2}}

What is the derivative of y=(x+1/x)/(2sqrt(x)+1/x) ?
The derivative of y=(x+1/x)/(2sqrt(x)+1/x) is (sqrt(x)(x^2+2sqrt(x)-3))/((2xsqrt(x)+1)^2)
What is the first derivative of y=(x+1/x)/(2sqrt(x)+1/x) ?
The first derivative of y=(x+1/x)/(2sqrt(x)+1/x) is (sqrt(x)(x^2+2sqrt(x)-3))/((2xsqrt(x)+1)^2)