What is (\partial)/(\partial x)(tan((xy)/(z^2))) ?

The answer to (\partial)/(\partial x)(tan((xy)/(z^2))) is (ysec^2(\frac{xy)/(z^2))}{z^2}

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(\partial)/(\partial x)(tan((xy)/(z^2)))

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

\frac{y sec ^{2} ( \frac{x y}{z ^{2}} )}{z ^{2}}

Solution steps

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

Treat

y, z

as constants

Apply the chain rule:

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

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

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

= sec^{2} (\frac{x y}{z ^{2}}) \frac{y}{z ^{2}}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{y sec ^{2} ( \frac{x y}{z ^{2}} )}{z ^{2}}

What is (\partial)/(\partial x)(tan((xy)/(z^2))) ?
The answer to (\partial)/(\partial x)(tan((xy)/(z^2))) is (ysec^2(\frac{xy)/(z^2))}{z^2}