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

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

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

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

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

Solution steps

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

Treat

y, z

as constants

Apply the chain rule:

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

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

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

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

Simplify

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

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

\int x^{3} sin (4 x) d x

\frac{5 x ^{4} - 3 x ^{3}}{x ^{5}}

\int_{0}^{1} \frac{4 - x}{1 + x ^{2}} d x

\int \frac{sec ^{2} ( 8 x )}{tan ( 8 x ) + 5} d x

\frac{\partial}{\partial x} (1 (x - y) e^{1 y + 5 x^{2}})

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