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

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

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

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

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

Solution steps

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

Treat

y, z

as constants

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

\frac{\partial}{\partial x} (sin (x z)) = cos (x z) z

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

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