What is (\partial)/(\partial x)(x^2y^3sin(x-2y)) ?

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

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(\partial)/(\partial x)(x^2y^3sin(x-2y))

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = x^{2}, g = sin (x - 2 y)

= y^{3} (\frac{\partial}{\partial x} (x^{2}) sin (x - 2 y) + \frac{\partial}{\partial x} (sin (x - 2 y)) x^{2})

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

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

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

\frac{d}{d x} (\frac{2 x cos ( x ^{2} )}{sin ( x ^{2} ) + 1})

\int \frac{x}{a ^{2} - x ^{2}} d x

x \to 25 lim (x)

f (x) = 7 e^{x}

\int \frac{x}{x ^{2} - 16} d x

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