What is the derivative of log_{2}(xlog_{5}(x)) ?

The derivative of log_{2}(xlog_{5}(x)) is (ln(x)+1)/(ln(2)xln(x))

What is the first derivative of log_{2}(xlog_{5}(x)) ?

The first derivative of log_{2}(xlog_{5}(x)) is (ln(x)+1)/(ln(2)xln(x))

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derivative of

lo g_{2} (x lo g_{5} (x))

\frac{ln ( x ) + 1}{ln ( 2 ) x ln ( x )}

Solution steps

\frac{d}{d x} (lo g_{2} (x lo g_{5} (x)))

Apply log rule:

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

= \frac{d}{d x} (\frac{ln ( x lo g _{5} ( x ) )}{ln ( 2 )})

Take the constant out:

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

= \frac{1}{ln ( 2 )} \frac{d}{d x} (ln (x lo g_{5} (x)))

Apply the chain rule:

\frac{1}{x lo g _{5} ( x )} \frac{d}{d x} (x lo g_{5} (x))

= \frac{1}{x lo g _{5} ( x )} \frac{d}{d x} (x lo g_{5} (x))

\frac{d}{d x} (x lo g_{5} (x)) = lo g_{5} (x) + \frac{1}{ln ( 5 )}

= \frac{1}{ln ( 2 )} \cdot \frac{1}{x lo g _{5} ( x )} (lo g_{5} (x) + \frac{1}{ln ( 5 )})

Simplify

\frac{1}{ln ( 2 )} \cdot \frac{1}{x lo g _{5} ( x )} (lo g_{5} (x) + \frac{1}{ln ( 5 )}) : \frac{ln ( x ) + 1}{ln ( 2 ) x ln ( x )}

= \frac{ln ( x ) + 1}{ln ( 2 ) x ln ( x )}

What is the derivative of log_{2}(xlog_{5}(x)) ?
The derivative of log_{2}(xlog_{5}(x)) is (ln(x)+1)/(ln(2)xln(x))
What is the first derivative of log_{2}(xlog_{5}(x)) ?
The first derivative of log_{2}(xlog_{5}(x)) is (ln(x)+1)/(ln(2)xln(x))