What is (\partial)/(\partial z)(z(y-yz)^4) ?

The answer to (\partial)/(\partial z)(z(y-yz)^4) is (y-zy)^4-4zy(y-zy)^3

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(\partial)/(\partial z)(z(y-yz)^4)

\frac{\partial}{\partial z} (z (y - yz)^{4})

(y - zy)^{4} - 4 zy (y - zy)^{3}

Solution steps

\frac{\partial}{\partial z} (z (y - yz)^{4})

Treat

y

as a constant

Apply the Product Rule:

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

f = z, g = (y - yz)^{4}

= \frac{\partial}{\partial z} (z) (y - yz)^{4} + \frac{\partial}{\partial z} ((y - yz)^{4}) z

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

\frac{\partial}{\partial z} ((y - yz)^{4}) = - 4 y (y - yz)^{3}

= 1 \cdot (y - yz)^{4} + (- 4 y (y - yz)^{3}) z

Simplify

= (y - zy)^{4} - 4 zy (y - zy)^{3}

\int_{0}^{5} e^{- x} d x

(x^{2} + 1) \frac{d y}{d x} + 3 x (y - 1) = 0, y (0) = 3

\frac{\partial}{\partial x} (x^{2} + y^{2} + \frac{2}{x y})

r^{2} - 16 x^{2}

\frac{\partial}{\partial x} (y - x^{2})

What is (\partial)/(\partial z)(z(y-yz)^4) ?
The answer to (\partial)/(\partial z)(z(y-yz)^4) is (y-zy)^4-4zy(y-zy)^3