What is (\partial)/(\partial y)(4x(4+y)^{-1}) ?

The answer to (\partial)/(\partial y)(4x(4+y)^{-1}) is -(4x)/((4+y)^2)

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(\partial)/(\partial y)(4x(4+y)^{-1})

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

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

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

\frac{\partial}{\partial y} (4 + y) = 1

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

Simplify

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

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

\frac{6 x}{x + 5}

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

\frac{d}{d x} (e^{\frac{2}{3} x})

What is (\partial)/(\partial y)(4x(4+y)^{-1}) ?
The answer to (\partial)/(\partial y)(4x(4+y)^{-1}) is -(4x)/((4+y)^2)