What is the derivative of (2x+6x^2/((1+6x)^2)) ?

The derivative of (2x+6x^2/((1+6x)^2)) is 2/((1+6x)^3)

What is the first derivative of (2x+6x^2/((1+6x)^2)) ?

The first derivative of (2x+6x^2/((1+6x)^2)) is 2/((1+6x)^3)

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\frac{d}{d x} (\frac{2 x + 6 x ^{2}}{( 1 + 6 x ) ^{2}})

\frac{2}{( 1 + 6 x ) ^{3}}

Solution steps

\frac{d}{d x} (\frac{2 x + 6 x ^{2}}{( 1 + 6 x ) ^{2}})

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( 2 x + 6 x ^{2} ) ( 1 + 6 x ) ^{2} - \frac{d}{d x} ( ( 1 + 6 x ) ^{2} ) ( 2 x + 6 x ^{2} )}{( ( 1 + 6 x ) ^{2} ) ^{2}}

\frac{d}{d x} (2 x + 6 x^{2}) = 2 + 12 x

\frac{d}{d x} ((1 + 6 x)^{2}) = 12 (1 + 6 x)

= \frac{( 2 + 12 x ) ( 1 + 6 x ) ^{2} - 12 ( 1 + 6 x ) ( 2 x + 6 x ^{2} )}{( ( 1 + 6 x ) ^{2} ) ^{2}}

Simplify

\frac{( 2 + 12 x ) ( 1 + 6 x ) ^{2} - 12 ( 1 + 6 x ) ( 2 x + 6 x ^{2} )}{( ( 1 + 6 x ) ^{2} ) ^{2}} : \frac{2}{( 1 + 6 x ) ^{3}}

= \frac{2}{( 1 + 6 x ) ^{3}}

y + x^{2} + y^{2} d x - x d y = 0

y^{^{''}} - 3 y^{^{'}} - 4 y = 0, y (0) = a, y^{^{'}} (0) = - 20

\frac{d}{d x} (x^{\frac{8}{3}} - 20 x^{\frac{2}{3}})

\frac{d y}{d x} = y^{2} sin (x^{2}), y (- 2) = \frac{1}{3}

f (r) = π r^{2}

What is the derivative of (2x+6x^2/((1+6x)^2)) ?
The derivative of (2x+6x^2/((1+6x)^2)) is 2/((1+6x)^3)
What is the first derivative of (2x+6x^2/((1+6x)^2)) ?
The first derivative of (2x+6x^2/((1+6x)^2)) is 2/((1+6x)^3)