What is the derivative of (9u^2)/((u^2+u)^3) ?

The derivative of (9u^2)/((u^2+u)^3) is (9(-4u-1))/(u^2(u+1)^4)

What is the first derivative of (9u^2)/((u^2+u)^3) ?

The first derivative of (9u^2)/((u^2+u)^3) is (9(-4u-1))/(u^2(u+1)^4)

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derivative

\frac{9 ( - 4 u - 1 )}{u ^{2} ( u + 1 ) ^{4}}

Solution steps

\frac{d}{d u} (\frac{9 u ^{2}}{( u ^{2} + u ) ^{3}})

Take the constant out:

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

= 9 \frac{d}{d u} (\frac{u ^{2}}{( u ^{2} + u ) ^{3}})

Apply the Quotient Rule:

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

= 9 \cdot \frac{\frac{d}{d u} ( u ^{2} ) ( u ^{2} + u ) ^{3} - \frac{d}{d u} ( ( u ^{2} + u ) ^{3} ) u ^{2}}{( ( u ^{2} + u ) ^{3} ) ^{2}}

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

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

= 9 \cdot \frac{2 u ( u ^{2} + u ) ^{3} - 3 ( u ^{2} + u ) ^{2} ( 2 u + 1 ) u ^{2}}{( ( u ^{2} + u ) ^{3} ) ^{2}}

Simplify

9 \cdot \frac{2 u ( u ^{2} + u ) ^{3} - 3 ( u ^{2} + u ) ^{2} ( 2 u + 1 ) u ^{2}}{( ( u ^{2} + u ) ^{3} ) ^{2}} : \frac{9 ( - 4 u - 1 )}{u ^{2} ( u + 1 ) ^{4}}

= \frac{9 ( - 4 u - 1 )}{u ^{2} ( u + 1 ) ^{4}}

What is the derivative of (9u^2)/((u^2+u)^3) ?
The derivative of (9u^2)/((u^2+u)^3) is (9(-4u-1))/(u^2(u+1)^4)
What is the first derivative of (9u^2)/((u^2+u)^3) ?
The first derivative of (9u^2)/((u^2+u)^3) is (9(-4u-1))/(u^2(u+1)^4)