What is the derivative of log_{10}(log_{10}(x)) ?

The derivative of log_{10}(log_{10}(x)) is 1/(ln^2(10)xlog_{10)(x)}

What is the first derivative of log_{10}(log_{10}(x)) ?

The first derivative of log_{10}(log_{10}(x)) is 1/(ln^2(10)xlog_{10)(x)}

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\frac{d}{d x} (lo g_{10} (lo g_{10} (x)))

\frac{1}{ln ^{2} ( 10 ) x lo g _{10} ( x )}

Solution steps

\frac{d}{d x} (lo g_{10} (lo g_{10} (x)))

Apply log rule:

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

= \frac{d}{d x} (\frac{ln ( lo g _{10} ( x ) )}{ln ( 10 )})

Take the constant out:

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

= \frac{1}{ln ( 10 )} \frac{d}{d x} (ln (lo g_{10} (x)))

Apply the chain rule:

\frac{1}{lo g _{10} ( x )} \frac{d}{d x} (lo g_{10} (x))

= \frac{1}{lo g _{10} ( x )} \frac{d}{d x} (lo g_{10} (x))

\frac{d}{d x} (lo g_{10} (x)) = \frac{1}{x ln ( 10 )}

= \frac{1}{ln ( 10 )} \cdot \frac{1}{lo g _{10} ( x )} \cdot \frac{1}{x ln ( 10 )}

Simplify

\frac{1}{ln ( 10 )} \cdot \frac{1}{lo g _{10} ( x )} \cdot \frac{1}{x ln ( 10 )} : \frac{1}{ln ^{2} ( 10 ) x lo g _{10} ( x )}

= \frac{1}{ln ^{2} ( 10 ) x lo g _{10} ( x )}

What is the derivative of log_{10}(log_{10}(x)) ?
The derivative of log_{10}(log_{10}(x)) is 1/(ln^2(10)xlog_{10)(x)}
What is the first derivative of log_{10}(log_{10}(x)) ?
The first derivative of log_{10}(log_{10}(x)) is 1/(ln^2(10)xlog_{10)(x)}