What is (\partial)/(\partial x)(10x^5y^4e^{-x}) ?

The answer to (\partial)/(\partial x)(10x^5y^4e^{-x}) is 10y^4(5e^{-x}x^4-e^{-x}x^5)

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(\partial)/(\partial x)(10x^5y^4e^{-x})

\frac{\partial}{\partial x} (10 x^{5} y^{4} e^{- x})

10 y^{4} (5 e^{- x} x^{4} - e^{- x} x^{5})

Solution steps

\frac{\partial}{\partial x} (10 x^{5} y^{4} e^{- x})

Treat

y

as a constant

Take the constant out:

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

= 10 y^{4} \frac{\partial}{\partial x} (x^{5} e^{- x})

Apply the Product Rule:

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

f = x^{5}, g = e^{- x}

= 10 y^{4} (\frac{\partial}{\partial x} (x^{5}) e^{- x} + \frac{\partial}{\partial x} (e^{- x}) x^{5})

\frac{\partial}{\partial x} (x^{5}) = 5 x^{4}

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

= 10 y^{4} (5 x^{4} e^{- x} + (- e^{- x}) x^{5})

Simplify

= 10 y^{4} (5 e^{- x} x^{4} - e^{- x} x^{5})

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What is (\partial)/(\partial x)(10x^5y^4e^{-x}) ?
The answer to (\partial)/(\partial x)(10x^5y^4e^{-x}) is 10y^4(5e^{-x}x^4-e^{-x}x^5)