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

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

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

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

\int t \cdot \frac{1}{e ^{t}} d t

\int (- 2 sec (x) tan (x) - 3 sec^{2} (x)) d x

5 t \cdot \frac{d y}{d t} + 3 y = t

\frac{d x}{d t} = \frac{( 2 t ^{2} + x ^{2} )}{t x}

y^{^{''}} - 2 y^{^{'}} + 5 y = e^{x} sin (x)

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