Derivative of ln ln ln x | Differentiate ln(ln(lnx))

The derivative of ln ln ln x is equal to 1/{x lnx ln(lnx)}. This is obtained by using the chain rule of differentiation. In this post, lets learn how to differentiate ln(ln(lnx)).

Differentiation of ln(ln(lnx))

Step 1:

Recall, the chain rule of differentiation: Suppose $f$ is a function of $z$ and $z$ is a function of $x$, that is, $f=f(z(x))$. By this rule, we have

$\dfrac{df}{dx}=\dfrac{df}{dz} \cdot \dfrac{dz}{dx}$

To find the derivative of ln(ln(lnx)), suppose z=ln(x).

Therefore, the derivative of ln(ln(lnx)) is equal to

Step 2:

$\dfrac{d}{dx}(\ln(\ln(\ln x)))$

= $\dfrac{d}{dx}(\ln(z))$ where $z=\ln(\ln x)$.

= $\dfrac{d}{dz}(\ln(z)) \cdot \dfrac{dz}{dx}$ using the chain rule of derivatives

Since the derivative of lnz is equal to 1/z, we obtain that

$\dfrac{d}{dx}(\ln(\ln(\ln x)))$ = $\dfrac{1}{z} \cdot \dfrac{dz}{dx}$ ….(A)

Step 3:

Now, $\dfrac{dz}{dx}$ = $\dfrac{d}{dx}(\ln(\ln x))$ = $\dfrac{1}{\ln x} \times \dfrac{d}{dx}(\ln x)$ = $\dfrac{1}{x\ln x}$.

Step 4:

Thus, using step 3 and step 4, it follows from the above Equation (A) that

$\dfrac{d}{dx}(\ln(\ln(\ln x)))$

= $\dfrac{1}{z} \cdot \dfrac{dz}{dx}$

= $\dfrac{1}{\ln(\ln x)} \cdot \dfrac{dz}{dx}$ as z=ln(lnx).

= $\dfrac{1}{\ln(\ln x)} \cdot \dfrac{1}{x\ln x}$

= $\dfrac{1}{x \ln(x) \ln(\ln x) }$

Therefore, the derivative of ln(ln(ln x)) is equal to $\dfrac{1}{x \ln(x) \ln(\ln x) }$, and this is obtained by the chain rule of differentiation.

Short Proof

$\dfrac{d}{dx}(\ln(\ln(\ln x)))$ = $\dfrac{1}{\ln(\ln x)} \times \dfrac{d}{dx}(\ln(\ln x))$, by chain rule. = $\dfrac{1}{\ln(\ln x)} \times \left\{ \dfrac{1}{\ln x} \times \dfrac{d}{dx}(\ln x) \right\}$, again by chain rule. = $\dfrac{1}{\ln(\ln x)} \times \dfrac{1}{\ln x} \times \dfrac{1}{x}$ = $\dfrac{1}{x \ln(x) \ln(\ln x) }$. So the derivative of ln(ln(ln x)) is equal to 1/{x lnx ln(lnx)}.

Also Read: Derivative of $\ln x$

Derivative of $\frac{1}{\ln x}$

Derivative of $\ln 2x$

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