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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

= - \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)

= - (\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) \cdot 2 y x

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

Simplify

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

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

n \to \infty lim (\frac{4}{n})

\frac{d}{d x} ((2 x + e^{x}) (5 - x))

\frac{2 x ^{2}}{x ^{2} - 1}

\frac{\partial}{\partial x} (e^{x^{2} y^{4}} sin (x yz))

4 (e^{x})^{9}

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