What is (\partial)/(\partial x)((5xyz)/(x^2+y^2+z^2)) ?

The answer to (\partial)/(\partial x)((5xyz)/(x^2+y^2+z^2)) is (5yz(-x^2+y^2+z^2))/((x^2+y^2+z^2)^2)

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(\partial)/(\partial x)((5xyz)/(x^2+y^2+z^2))

\frac{\partial}{\partial x} (\frac{5 x yz}{x ^{2} + y ^{2} + z ^{2}})

\frac{5 yz ( - x ^{2} + y ^{2} + z ^{2} )}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{5 x yz}{x ^{2} + y ^{2} + z ^{2}})

Treat

y, z

as constants

Take the constant out:

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

= 5 yz \frac{\partial}{\partial x} (\frac{x}{x ^{2} + y ^{2} + z ^{2}})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

= 5 yz \frac{1 \cdot ( x ^{2} + y ^{2} + z ^{2} ) - 2 xx}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

Simplify

5 yz \frac{1 \cdot ( x ^{2} + y ^{2} + z ^{2} ) - 2 xx}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}} : \frac{5 yz ( - x ^{2} + y ^{2} + z ^{2} )}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

= \frac{5 yz ( - x ^{2} + y ^{2} + z ^{2} )}{( x ^{2} + y ^{2} + z ^{2} ) ^{2}}

y^{^{'}} + 2 y = 10 sin (t)

d er i v a t i v e e^{t} \cdot sin (2 t)

x \to 1 lim (\frac{x ^{2 n} - 2 x ^{n} + 1}{x ^{2 m} - 2 x ^{m} + 1})

\frac{d P}{d t} = Pt, P (1) = 2

\int_{0}^{\frac{π}{4}} tan^{4} (t) d t

What is (\partial)/(\partial x)((5xyz)/(x^2+y^2+z^2)) ?
The answer to (\partial)/(\partial x)((5xyz)/(x^2+y^2+z^2)) is (5yz(-x^2+y^2+z^2))/((x^2+y^2+z^2)^2)