What is (\partial)/(\partial y)(xyz^2tan(yt)) ?

The answer to (\partial)/(\partial y)(xyz^2tan(yt)) is xz^2(tan(yt)+ytsec^2(yt))

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(\partial)/(\partial y)(xyz^2tan(yt))

\frac{\partial}{\partial y} (x y z^{2} tan (y t))

x z^{2} (tan (y t) + y t sec^{2} (y t))

Solution steps

\frac{\partial}{\partial y} (x y z^{2} tan (y t))

Treat

x, z, t

as constants

Take the constant out:

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

= x z^{2} \frac{\partial}{\partial y} (y tan (y t))

Apply the Product Rule:

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

f = y, g = tan (y t)

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

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

\frac{\partial}{\partial y} (tan (y t)) = sec^{2} (y t) t

= x z^{2} (1 \cdot tan (y t) + sec^{2} (y t) t y)

Simplify

= x z^{2} (tan (y t) + y t sec^{2} (y t))

y = 2 - 6 x, at x = - 1

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What is (\partial)/(\partial y)(xyz^2tan(yt)) ?
The answer to (\partial)/(\partial y)(xyz^2tan(yt)) is xz^2(tan(yt)+ytsec^2(yt))