What is (\partial)/(\partial x)(xcos(x/y)) ?

The answer to (\partial)/(\partial x)(xcos(x/y)) is cos(x/y)-(xsin(x/y))/y

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(\partial)/(\partial x)(xcos(x/y))

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

cos (\frac{x}{y}) - \frac{x sin ( \frac{x}{y} )}{y}

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = cos (\frac{x}{y})

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

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

\frac{\partial}{\partial x} (cos (\frac{x}{y})) = - \frac{sin ( \frac{x}{y} )}{y}

= 1 \cdot cos (\frac{x}{y}) + - \frac{sin ( \frac{x}{y} )}{y} x

Simplify

1 \cdot cos (\frac{x}{y}) + - \frac{sin ( \frac{x}{y} )}{y} x : cos (\frac{x}{y}) - \frac{x sin ( \frac{x}{y} )}{y}

= cos (\frac{x}{y}) - \frac{x sin ( \frac{x}{y} )}{y}

x \to 0 lim (x^{3} - 2 x^{3} - 4 x + 7)

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

\frac{d}{d x} (x^{2} + 4 y^{2} - 4 x y + 2)

f (x) = (2 x + 1)^{5} + 10 x (2 x + 1)^{4}

y^{^{'}} + \frac{y ^{2}}{x ^{2}} = 0

What is (\partial)/(\partial x)(xcos(x/y)) ?
The answer to (\partial)/(\partial x)(xcos(x/y)) is cos(x/y)-(xsin(x/y))/y