What is (\partial)/(\partial z)(zsin(xy^2+2z)) ?

The answer to (\partial)/(\partial z)(zsin(xy^2+2z)) is sin(xy^2+2z)+2zcos(xy^2+2z)

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(\partial)/(\partial z)(zsin(xy^2+2z))

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

sin (x y^{2} + 2 z) + 2 z cos (x y^{2} + 2 z)

Solution steps

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

Treat

x, y

as constants

Apply the Product Rule:

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

f = z, g = sin (x y^{2} + 2 z)

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

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

\frac{\partial}{\partial z} (sin (x y^{2} + 2 z)) = cos (x y^{2} + 2 z) \cdot 2

= 1 \cdot sin (x y^{2} + 2 z) + cos (x y^{2} + 2 z) \cdot 2 z

Simplify

= sin (x y^{2} + 2 z) + 2 z cos (x y^{2} + 2 z)

What is (\partial)/(\partial z)(zsin(xy^2+2z)) ?
The answer to (\partial)/(\partial z)(zsin(xy^2+2z)) is sin(xy^2+2z)+2zcos(xy^2+2z)