What is (\partial)/(\partial y)(|y|+1/(2a)|x-y|^2) ?

The answer to (\partial)/(\partial y)(|y|+1/(2a)|x-y|^2) is y/(|y|)-(x-y)/a

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(\partial)/(\partial y)(|y|+1/(2a)|x-y|^2)

\frac{\partial}{\partial y} (∣ y ∣ + \frac{1}{2 a} ∣ x - y ∣^{2})

\frac{y}{∣ y ∣} - \frac{x - y}{a}

Solution steps

\frac{\partial}{\partial y} (∣ y ∣ + \frac{1}{2 a} ∣ x - y ∣^{2})

Simplify

∣ y ∣ + \frac{1}{2 a} ∣ x - y ∣^{2} : ∣ y ∣ + \frac{( x - y ) ^{2}}{2 a}

= \frac{\partial}{\partial y} (∣ y ∣ + \frac{( x - y ) ^{2}}{2 a})

Treat

a, x

as a constant

Apply the Sum/Difference Rule:

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

= \frac{\partial}{\partial y} (∣ y ∣) + \frac{\partial}{\partial y} (\frac{( x - y ) ^{2}}{2 a})

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

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

= \frac{y}{∣ y ∣} - \frac{x - y}{a}

\int \frac{1}{4 x + 1} d x

\frac{\partial}{\partial x} (\frac{7}{x})

x \to 4 + lim (\frac{x + 1}{x - 4})

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

n = 1 \sum \infty \frac{n}{( n ^{3} + 1 )}

What is (\partial)/(\partial y)(|y|+1/(2a)|x-y|^2) ?
The answer to (\partial)/(\partial y)(|y|+1/(2a)|x-y|^2) is y/(|y|)-(x-y)/a