What is (\partial)/(\partial x)(x^3cos(xy)) ?

The answer to (\partial)/(\partial x)(x^3cos(xy)) is 3x^2cos(xy)-x^3ysin(xy)

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(\partial)/(\partial x)(x^3cos(xy))

\frac{\partial}{\partial x} (x^{3} cos (x y))

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

Solution steps

\frac{\partial}{\partial x} (x^{3} cos (x y))

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

x = y^{2}, x = 6 - 2 y^{2}

\int (9 x 1 + 2 x^{2}) d x

x \to 0 lim (\frac{x ^{2} - x}{x})

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

\int_{0}^{1} x - x^{4} d x

What is (\partial)/(\partial x)(x^3cos(xy)) ?
The answer to (\partial)/(\partial x)(x^3cos(xy)) is 3x^2cos(xy)-x^3ysin(xy)