What is (\partial)/(\partial x)(sqrt((a+x)^2+y^2)) ?

The answer to (\partial)/(\partial x)(sqrt((a+x)^2+y^2)) is (x+a)/(sqrt((a+x)^2+y^2))

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(\partial)/(\partial x)(sqrt((a+x)^2+y^2))

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

\frac{x + a}{( a + x ) ^{2} + y ^{2}}

Solution steps

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

Treat

a, y

as a constant

Apply the chain rule:

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

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

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

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

Simplify

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

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

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