What is the derivative of f(x)=(4x^2)/((x^2+x)^3) ?

The derivative of f(x)=(4x^2)/((x^2+x)^3) is (4(-4x-1))/(x^2(x+1)^4)

What is the first derivative of f(x)=(4x^2)/((x^2+x)^3) ?

The first derivative of f(x)=(4x^2)/((x^2+x)^3) is (4(-4x-1))/(x^2(x+1)^4)

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derivative of

f (x) = \frac{4 x ^{2}}{( x ^{2} + x ) ^{3}}

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

Solution steps

\frac{d}{d x} (\frac{4 x ^{2}}{( x ^{2} + x ) ^{3}})

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= 4 \frac{d}{d x} (\frac{x ^{2}}{( x ^{2} + x ) ^{3}})

Apply the Quotient Rule:

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

= 4 \cdot \frac{\frac{d}{d x} ( x ^{2} ) ( x ^{2} + x ) ^{3} - \frac{d}{d x} ( ( x ^{2} + x ) ^{3} ) x ^{2}}{( ( x ^{2} + x ) ^{3} ) ^{2}}

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

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

= 4 \cdot \frac{2 x ( x ^{2} + x ) ^{3} - 3 ( x ^{2} + x ) ^{2} ( 2 x + 1 ) x ^{2}}{( ( x ^{2} + x ) ^{3} ) ^{2}}

Simplify

4 \cdot \frac{2 x ( x ^{2} + x ) ^{3} - 3 ( x ^{2} + x ) ^{2} ( 2 x + 1 ) x ^{2}}{( ( x ^{2} + x ) ^{3} ) ^{2}} : \frac{4 ( - 4 x - 1 )}{x ^{2} ( x + 1 ) ^{4}}

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

What is the derivative of f(x)=(4x^2)/((x^2+x)^3) ?
The derivative of f(x)=(4x^2)/((x^2+x)^3) is (4(-4x-1))/(x^2(x+1)^4)
What is the first derivative of f(x)=(4x^2)/((x^2+x)^3) ?
The first derivative of f(x)=(4x^2)/((x^2+x)^3) is (4(-4x-1))/(x^2(x+1)^4)