What is (\partial)/(\partial x)(I/(sqrt(x*y))) ?

The answer to (\partial)/(\partial x)(I/(sqrt(x*y))) is -I/(2x^{3/2)y^{1/2}}

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(\partial)/(\partial x)(I/(sqrt(x*y)))

\frac{\partial}{\partial x} (\frac{I}{x \cdot y})

- \frac{I}{2 x ^{\frac{3}{2}} y ^{\frac{1}{2}}}

Solution steps

\frac{\partial}{\partial x} (\frac{I}{x y})

Treat

I, y

as constants

Take the constant out:

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

= I \frac{\partial}{\partial x} (\frac{1}{x y})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= I \frac{\partial}{\partial x} ((x y)^{- \frac{1}{2}})

Apply the chain rule:

- \frac{1}{2 ( x y ) ^{\frac{3}{2}}} \frac{\partial}{\partial x} (x y)

= - \frac{1}{2 ( x y ) ^{\frac{3}{2}}} \frac{\partial}{\partial x} (x y)

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

= I (- \frac{1}{2 ( x y ) ^{\frac{3}{2}}} y)

Simplify

I (- \frac{1}{2 ( x y ) ^{\frac{3}{2}}} y) : - \frac{I}{2 x ^{\frac{3}{2}} y ^{\frac{1}{2}}}

= - \frac{I}{2 x ^{\frac{3}{2}} y ^{\frac{1}{2}}}

5 y^{^{''}} + y^{^{'}} = - 10 x

\frac{4 x}{x + 2}

\int x e^{x - 2} d x

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

x \to 0 lim (\frac{3 x}{2 x ^{2} - x})

What is (\partial)/(\partial x)(I/(sqrt(x*y))) ?
The answer to (\partial)/(\partial x)(I/(sqrt(x*y))) is -I/(2x^{3/2)y^{1/2}}