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}}

\int \frac{5}{cos ^{2} ( x )} d x

\frac{d ^{2} y}{d x ^{2}} + 14 \frac{d y}{d x} + 98 y = 0

(3 x y^{2} + 1 y) d x + (3 x^{2} y + 1 x) d y = 0

\int cos (3 x) sin^{4} (3 x) d x

\frac{d y}{d x} = (4 x - 1) (y + 5)

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))