What is (\partial)/(\partial y)(e^{xyz^5}yz^5) ?

The answer to (\partial)/(\partial y)(e^{xyz^5}yz^5) is z^5(e^{xyz^5}xyz^5+e^{xyz^5})

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(\partial)/(\partial y)(e^{xyz^5}yz^5)

\frac{\partial}{\partial y} (e^{x y z^{5}} y z^{5})

z^{5} (e^{x y z^{5}} x y z^{5} + e^{x y z^{5}})

Solution steps

\frac{\partial}{\partial y} (e^{x y z^{5}} y z^{5})

Treat

x, z

as constants

Take the constant out:

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

= z^{5} \frac{\partial}{\partial y} (e^{x y z^{5}} y)

Apply the Product Rule:

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

f = e^{x y z^{5}}, g = y

= z^{5} (\frac{\partial}{\partial y} (e^{x y z^{5}}) y + \frac{\partial}{\partial y} (y) e^{x y z^{5}})

\frac{\partial}{\partial y} (e^{x y z^{5}}) = e^{x y z^{5}} x z^{5}

\frac{\partial}{\partial y} (y) = 1

= z^{5} (e^{x y z^{5}} x z^{5} y + 1 \cdot e^{x y z^{5}})

Simplify

= z^{5} (e^{x y z^{5}} x y z^{5} + e^{x y z^{5}})

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

f (x) = x^{2} + x + 9, at x = 2

\int 5 e^{3 x} d x

h (w) = \frac{( x + 1 ) e ^{x}}{w - 2}

f (x) = 5 x^{2} - 5 x

What is (\partial)/(\partial y)(e^{xyz^5}yz^5) ?
The answer to (\partial)/(\partial y)(e^{xyz^5}yz^5) is z^5(e^{xyz^5}xyz^5+e^{xyz^5})