What is the derivative of (x+2x^2/(1+x^2)) ?

The derivative of (x+2x^2/(1+x^2)) is (-x^2+4x+1)/((1+x^2)^2)

What is the first derivative of (x+2x^2/(1+x^2)) ?

The first derivative of (x+2x^2/(1+x^2)) is (-x^2+4x+1)/((1+x^2)^2)

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

\frac{- x ^{2} + 4 x + 1}{( 1 + x ^{2} ) ^{2}}

Solution steps

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

Apply the Quotient Rule:

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

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

\frac{d}{d x} (x + 2 x^{2}) = 1 + 4 x

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

= \frac{( 1 + 4 x ) ( 1 + x ^{2} ) - 2 x ( x + 2 x ^{2} )}{( 1 + x ^{2} ) ^{2}}

Expand

(1 + 4 x) (1 + x^{2}) - 2 x (x + 2 x^{2}) : - x^{2} + 4 x + 1

= \frac{- x ^{2} + 4 x + 1}{( 1 + x ^{2} ) ^{2}}

x \to 4 - lim (\frac{3}{x - 2})

\int (x + \frac{1}{5 x}) d x

y = \frac{1}{12 x ^{2}}, y = x, y = \frac{x}{18}

\int \frac{3 x ^{5} - 6 x ^{2} + 2 x - 5}{x ^{3}} d x

n = 0 \sum \infty e^{4 n}

What is the derivative of (x+2x^2/(1+x^2)) ?
The derivative of (x+2x^2/(1+x^2)) is (-x^2+4x+1)/((1+x^2)^2)
What is the first derivative of (x+2x^2/(1+x^2)) ?
The first derivative of (x+2x^2/(1+x^2)) is (-x^2+4x+1)/((1+x^2)^2)