What is (\partial)/(\partial x)(x^{-3}y^2+xy^2+5xy) ?

The answer to (\partial)/(\partial x)(x^{-3}y^2+xy^2+5xy) is -(3y^2)/(x^4)+y^2+5y

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(\partial)/(\partial x)(x^{-3}y^2+xy^2+5xy)

\frac{\partial}{\partial x} (x^{- 3} y^{2} + x y^{2} + 5 x y)

- \frac{3 y ^{2}}{x ^{4}} + y^{2} + 5 y

Solution steps

\frac{\partial}{\partial x} (x^{- 3} y^{2} + x y^{2} + 5 x y)

Treat

y

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= \frac{\partial}{\partial x} (x^{- 3} y^{2}) + \frac{\partial}{\partial x} (x y^{2}) + \frac{\partial}{\partial x} (5 x y)

\frac{\partial}{\partial x} (x^{- 3} y^{2}) = - \frac{3 y ^{2}}{x ^{4}}

\frac{\partial}{\partial x} (x y^{2}) = y^{2}

\frac{\partial}{\partial x} (5 x y) = 5 y

= - \frac{3 y ^{2}}{x ^{4}} + y^{2} + 5 y

\int \frac{1}{x - a} d x

\frac{8}{( z ^{2} + 4 ) ^{2}}

\int (x^{2} + 8 x) cos (x) d x

\frac{d}{d x} (x^{4})

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

What is (\partial)/(\partial x)(x^{-3}y^2+xy^2+5xy) ?
The answer to (\partial)/(\partial x)(x^{-3}y^2+xy^2+5xy) is -(3y^2)/(x^4)+y^2+5y