What is (\partial)/(\partial x)(x/(y^2)sin(x/y)) ?

The answer to (\partial)/(\partial x)(x/(y^2)sin(x/y)) is (ysin(x/y)+xcos(x/y))/(y^3)

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(\partial)/(\partial x)(x/(y^2)sin(x/y))

\frac{\partial}{\partial x} (\frac{x}{y ^{2}} sin (\frac{x}{y}))

\frac{y sin ( \frac{x}{y} ) + x cos ( \frac{x}{y} )}{y ^{3}}

Solution steps

\frac{\partial}{\partial x} (\frac{x}{y ^{2}} sin (\frac{x}{y}))

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

\frac{1}{y ^{2}} 1 \cdot sin (\frac{x}{y}) + \frac{cos ( \frac{x}{y} )}{y} x : \frac{y sin ( \frac{x}{y} ) + x cos ( \frac{x}{y} )}{y ^{3}}

= \frac{y sin ( \frac{x}{y} ) + x cos ( \frac{x}{y} )}{y ^{3}}

\int_{1}^{4} 7 t ln (t) d t

y = (7 ln (x))^{2}

\frac{\partial}{\partial x} (x^{e})

\int_{- \infty}^{0} z e^{9 z} d z

\int \frac{x}{( x + t ) ^{2}}

What is (\partial)/(\partial x)(x/(y^2)sin(x/y)) ?
The answer to (\partial)/(\partial x)(x/(y^2)sin(x/y)) is (ysin(x/y)+xcos(x/y))/(y^3)