What is the derivative of f(u)= u/(u+a/u) ?

The derivative of f(u)= u/(u+a/u) is (2au)/((u^2+a)^2)

What is the first derivative of f(u)= u/(u+a/u) ?

The first derivative of f(u)= u/(u+a/u) is (2au)/((u^2+a)^2)

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derivative of

f (u) = \frac{u}{u + \frac{a}{u}}

\frac{2 a u}{( u ^{2} + a ) ^{2}}

Solution steps

\frac{d}{d u} (\frac{u}{u + \frac{a}{u}})

Apply the Quotient Rule:

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

= \frac{\frac{d u}{d u} ( u + \frac{a}{u} ) - \frac{d}{d u} ( u + \frac{a}{u} ) u}{( u + \frac{a}{u} ) ^{2}}

\frac{d u}{d u} = 1

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

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

Simplify

\frac{1 \cdot ( u + \frac{a}{u} ) - ( 1 - \frac{a}{u ^{2}} ) u}{( u + \frac{a}{u} ) ^{2}} : \frac{2 a u}{( u ^{2} + a ) ^{2}}

= \frac{2 a u}{( u ^{2} + a ) ^{2}}

What is the derivative of f(u)= u/(u+a/u) ?
The derivative of f(u)= u/(u+a/u) is (2au)/((u^2+a)^2)
What is the first derivative of f(u)= u/(u+a/u) ?
The first derivative of f(u)= u/(u+a/u) is (2au)/((u^2+a)^2)