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

The answer to (\partial)/(\partial y)(x^nsin(y/(x^2))) is x^{n-2}cos(y/(x^2))

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

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

x^{n - 2} cos (\frac{y}{x ^{2}})

Solution steps

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

Treat

n, x

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

= x^{n} cos (\frac{y}{x ^{2}}) \frac{1}{x ^{2}}

Simplify

x^{n} cos (\frac{y}{x ^{2}}) \frac{1}{x ^{2}} : x^{n - 2} cos (\frac{y}{x ^{2}})

= x^{n - 2} cos (\frac{y}{x ^{2}})

f (x) = e^{3 t s i n (2 t)}

\int \frac{3}{u ^{2} + 4} d u

x \to - 1 lim (\frac{e ^{x}}{1 + x})

\int (x^{2} - 1) x d x

\frac{\partial}{\partial x} (- 1 - 2 x)

What is (\partial)/(\partial y)(x^nsin(y/(x^2))) ?
The answer to (\partial)/(\partial y)(x^nsin(y/(x^2))) is x^{n-2}cos(y/(x^2))