What is (\partial)/(\partial y)(e^{xyz^7}yz^7) ?

The answer to (\partial)/(\partial y)(e^{xyz^7}yz^7) is z^7(e^{xyz^7}xyz^7+e^{xyz^7})

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(\partial)/(\partial y)(e^{xyz^7}yz^7)

\frac{\partial}{\partial y} (e^{x y z^{7}} y z^{7})

z^{7} (e^{x y z^{7}} x y z^{7} + e^{x y z^{7}})

Solution steps

\frac{\partial}{\partial y} (e^{x y z^{7}} y z^{7})

Treat

x, z

as constants

Take the constant out:

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

= z^{7} \frac{\partial}{\partial y} (e^{x y z^{7}} y)

Apply the Product Rule:

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

f = e^{x y z^{7}}, g = y

= z^{7} (\frac{\partial}{\partial y} (e^{x y z^{7}}) y + \frac{\partial}{\partial y} (y) e^{x y z^{7}})

\frac{\partial}{\partial y} (e^{x y z^{7}}) = e^{x y z^{7}} x z^{7}

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

= z^{7} (e^{x y z^{7}} x z^{7} y + 1 \cdot e^{x y z^{7}})

Simplify

= z^{7} (e^{x y z^{7}} x y z^{7} + e^{x y z^{7}})

\frac{d v}{d t} = k (250 - v)

(x^{2} + 1)^{3} (2 x^{3} - 5 x)^{2}

\frac{d}{d x} (\frac{1 + x}{x})

\int \frac{2 x - 3}{1 - x ^{2}} d x

y^{^{''}} + y - 2 \cdot cos (x) = 0

What is (\partial)/(\partial y)(e^{xyz^7}yz^7) ?
The answer to (\partial)/(\partial y)(e^{xyz^7}yz^7) is z^7(e^{xyz^7}xyz^7+e^{xyz^7})