What is (\partial)/(\partial x)(log_{10}(xy)) ?

The answer to (\partial)/(\partial x)(log_{10}(xy)) is 1/(ln(10)x)

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(\partial)/(\partial x)(log_{10}(xy))

\frac{\partial}{\partial x} (lo g_{10} (x y))

\frac{1}{ln ( 10 ) x}

Solution steps

\frac{\partial}{\partial x} (lo g_{10} (x y))

Treat

y

as a constant

Apply log rule:

lo g_{a} (b) = \frac{ln ( b )}{ln ( a )}

= \frac{\partial}{\partial x} (\frac{ln ( x y )}{ln ( 10 )})

Take the constant out:

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

= \frac{1}{ln ( 10 )} \frac{\partial}{\partial x} (ln (x y))

Apply the chain rule:

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

= \frac{1}{x y} \frac{\partial}{\partial x} (x y)

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

= \frac{1}{ln ( 10 )} \cdot \frac{1}{x y} y

Simplify

\frac{1}{ln ( 10 )} \cdot \frac{1}{x y} y : \frac{1}{ln ( 10 ) x}

= \frac{1}{ln ( 10 ) x}

What is (\partial)/(\partial x)(log_{10}(xy)) ?
The answer to (\partial)/(\partial x)(log_{10}(xy)) is 1/(ln(10)x)