What is (\partial)/(\partial y)(xln(5x+8y)) ?

The answer to (\partial)/(\partial y)(xln(5x+8y)) is (8x)/(5x+8y)

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(\partial)/(\partial y)(xln(5x+8y))

\frac{\partial}{\partial y} (x ln (5 x + 8 y))

\frac{8 x}{5 x + 8 y}

Solution steps

\frac{\partial}{\partial y} (x ln (5 x + 8 y))

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= x \frac{\partial}{\partial y} (ln (5 x + 8 y))

Apply the chain rule:

\frac{1}{5 x + 8 y} \frac{\partial}{\partial y} (5 x + 8 y)

= \frac{1}{5 x + 8 y} \frac{\partial}{\partial y} (5 x + 8 y)

\frac{\partial}{\partial y} (5 x + 8 y) = 8

= x \frac{1}{5 x + 8 y} \cdot 8

Simplify

x \frac{1}{5 x + 8 y} \cdot 8 : \frac{8 x}{5 x + 8 y}

= \frac{8 x}{5 x + 8 y}

What is (\partial)/(\partial y)(xln(5x+8y)) ?
The answer to (\partial)/(\partial y)(xln(5x+8y)) is (8x)/(5x+8y)