What is (\partial)/(\partial x)(x^3y^4sin^2(xy)) ?

The answer to (\partial)/(\partial x)(x^3y^4sin^2(xy)) is y^4(3x^2sin^2(xy)+ysin(2yx)x^3)

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(\partial)/(\partial x)(x^3y^4sin^2(xy))

\frac{\partial}{\partial x} (x^{3} y^{4} sin^{2} (x y))

y^{4} (3 x^{2} sin^{2} (x y) + y sin (2 y x) x^{3})

Solution steps

\frac{\partial}{\partial x} (x^{3} y^{4} sin^{2} (x y))

Treat

y

as a constant

Take the constant out:

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

= y^{4} \frac{\partial}{\partial x} (x^{3} sin^{2} (x y))

Apply the Product Rule:

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

f = x^{3}, g = sin^{2} (x y)

= y^{4} (\frac{\partial}{\partial x} (x^{3}) sin^{2} (x y) + \frac{\partial}{\partial x} (sin^{2} (x y)) x^{3})

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

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

= y^{4} (3 x^{2} sin^{2} (x y) + y sin (2 y x) x^{3})

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What is (\partial)/(\partial x)(x^3y^4sin^2(xy)) ?
The answer to (\partial)/(\partial x)(x^3y^4sin^2(xy)) is y^4(3x^2sin^2(xy)+ysin(2yx)x^3)