What is (\partial)/(\partial x)(x^3e^ysin(x-y)) ?

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

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(\partial)/(\partial x)(x^3e^ysin(x-y))

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

e^{y} (3 x^{2} sin (x - y) + cos (x - y) x^{3})

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = x^{3}, g = sin (x - y)

= e^{y} (\frac{\partial}{\partial x} (x^{3}) sin (x - y) + \frac{\partial}{\partial x} (sin (x - y)) x^{3})

\frac{\partial}{\partial x} (x^{3}) = 3 x^{2}

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

= e^{y} (3 x^{2} sin (x - y) + cos (x - y) x^{3})

t an g e n t f (x) = x^{2} - 3, (3, 6)

ma c l a u r in \frac{4 x}{x ^{2} + 2 x - 3}

\frac{d}{d x} (\frac{1}{5 - 11 e ^{x}})

\int \frac{2}{3} x^{\frac{1}{2}} d x

\int \frac{x ^{3}}{e ^{x^{2}}} d x

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