What is (\partial)/(\partial x)(z^3x^2e^{zx}+x^2z) ?

The answer to (\partial)/(\partial x)(z^3x^2e^{zx}+x^2z) is z^3(2xe^{zx}+e^{zx}zx^2)+2zx

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(\partial)/(\partial x)(z^3x^2e^{zx}+x^2z)

\frac{\partial}{\partial x} (z^{3} x^{2} e^{z x} + x^{2} z)

z^{3} (2 x e^{z x} + e^{z x} z x^{2}) + 2 z x

Solution steps

\frac{\partial}{\partial x} (z^{3} x^{2} e^{z x} + x^{2} z)

Treat

z

as a constant

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

= z^{3} (2 x e^{z x} + e^{z x} z x^{2}) + 2 z x

What is (\partial)/(\partial x)(z^3x^2e^{zx}+x^2z) ?
The answer to (\partial)/(\partial x)(z^3x^2e^{zx}+x^2z) is z^3(2xe^{zx}+e^{zx}zx^2)+2zx