What is (\partial)/(\partial y)(ln(1+x^2+e^y+z)) ?

The answer to (\partial)/(\partial y)(ln(1+x^2+e^y+z)) is (e^y)/(1+x^2+e^y+z)

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(\partial)/(\partial y)(ln(1+x^2+e^y+z))

\frac{\partial}{\partial y} (ln (1 + x^{2} + e^{y} + z))

\frac{e ^{y}}{1 + x ^{2} + e ^{y} + z}

Solution steps

\frac{\partial}{\partial y} (ln (1 + x^{2} + e^{y} + z))

Treat

x, z

as constants

Apply the chain rule:

\frac{1}{1 + x ^{2} + e ^{y} + z} \frac{\partial}{\partial y} (1 + x^{2} + e^{y} + z)

= \frac{1}{1 + x ^{2} + e ^{y} + z} \frac{\partial}{\partial y} (1 + x^{2} + e^{y} + z)

\frac{\partial}{\partial y} (1 + x^{2} + e^{y} + z) = e^{y}

= \frac{1}{1 + x ^{2} + e ^{y} + z} e^{y}

Simplify

\frac{1}{1 + x ^{2} + e ^{y} + z} e^{y} : \frac{e ^{y}}{1 + x ^{2} + e ^{y} + z}

= \frac{e ^{y}}{1 + x ^{2} + e ^{y} + z}

\frac{s ^{2} + 6 s + 2}{s ( s + 1 ) ^{2}}

y^{^{'}} = t y - 2 t + y - 2

3 t^{3} - 2 t^{2} - 7 t + 2

\frac{\partial}{\partial x} (1^{x})

\frac{( x ^{2} + 11 )}{x ^{3}}

What is (\partial)/(\partial y)(ln(1+x^2+e^y+z)) ?
The answer to (\partial)/(\partial y)(ln(1+x^2+e^y+z)) is (e^y)/(1+x^2+e^y+z)