Solve w=e^x+x\ln{y}+y\ln{x}

Simplify

w = e^{x} + x ln (y) + y ln (x)

Find the first partial derivative by treating the variable y as a constant and differentiating with respect to x

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

Use differentiation rule \frac{\partial}{\partial x} (f (x) \pm g (x)) = \frac{\partial}{\partial x} (f (x)) \pm \frac{\partial}{\partial x} (g (x))

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

Use \frac{\partial}{\partial x} e^{x} = e^{x} to find derivative

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

Evaluate

More Steps

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

Evaluate

More Steps

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

Reduce fractions to a common denominator

\frac{\partial w}{\partial x} = \frac{e ^{x} x}{x} + \frac{ln ( y ) \times x}{x} + \frac{y}{x}

Reorder the terms

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

Solution

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