What is (\partial)/(\partial x)(x^3y^2e^{xz}) ?

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

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(\partial)/(\partial x)(x^3y^2e^{xz})

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

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

Solution steps

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

Treat

y, z

as constants

Take the constant out:

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

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

Apply the Product Rule:

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

f = x^{3}, g = e^{x z}

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

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

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

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

\frac{d}{d x} (\frac{1 + e ^{- 2 x}}{2 + cos ( 2 x )})

\int (cos (\frac{x}{2}))^{2} d x

f (x) = 3 x^{3} - x^{2} + 1, (2, 21)

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

\int (2 x)^{2} d x

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