What is the Derivative of ln(lnx)

The derivative of ln(lnx) is equal to 1/xln(x) and it is denoted by d/dx (x lnx). So the derivative formula of ln(lnx) is given by

$\dfrac{d}{dx}\big(\ln(\ln x) \big) = \dfrac{1}{x \ln x}$.

Derivative of ln(lnx) by Chain Rule

Answer: The derivative of ln(lnx) is $\dfrac{1}{x \ln x}$.

Explanation:

Let z=ln x.

Differentiating, $\dfrac{dz}{dx}=\dfrac{1}{x}$.

Now, $\dfrac{d}{dx}\big(\ln(\ln x) \big)$

= $\dfrac{d}{dz}\big(\ln z\big) \times \dfrac{dz}{dx}$ by the chain rule of derivatives.

= $\dfrac{1}{z} \times \dfrac{1}{x}$ as dz/dx=1/x.

= $\dfrac{1}{zx}$

= $\dfrac{1}{x \ln x}$ as we have z=lnx.

So the derivative of ln(ln x) is equal to 1/xlnx, and it is obtained by the chain rule of derivatives.

More Derivatives:

Derivative of 1/lnx | Derivative of ln(1/x)

Derivative of 1/sinx | Derivative of 1/cosx

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