What is (\partial)/(\partial x)(e^{ye^x}) ?

The answer to (\partial)/(\partial x)(e^{ye^x}) is e^{e^xy+x}y

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(\partial)/(\partial x)(e^{ye^x})

\frac{\partial}{\partial x} (e^{y e^{x}})

e^{e^{x} y + x} y

Solution steps

\frac{\partial}{\partial x} (e^{y e^{x}})

Treat

y

as a constant

Apply the chain rule:

e^{y e^{x}} \frac{\partial}{\partial x} (y e^{x})

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

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

= e^{y e^{x}} y e^{x}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{y e^{x}} e^{x} = e^{e^{x} y + x}

= e^{e^{x} y + x} y

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What is (\partial)/(\partial x)(e^{ye^x}) ?
The answer to (\partial)/(\partial x)(e^{ye^x}) is e^{e^xy+x}y