Using the differentiation rules and the derivative table, find the derivative of the function y=\ln\ln x


y' = \frac{1}{x \ln(x)}

We can use the Chain Rule. Recall that if f and g are two differentiable functions, then the composition f\circ g is differentiable and its derivative in x is given by

[f\circ g]'(x)=f'(g(x))\cdot g'(x)

  • Identify the outer function and the inner function
    We can rewrite y as


    y = \ln(g(x))


    namely the composition f\circ g, where g(x) = \ln(x) and the outer function is f(\cdot)=\ln(\cdot)

  • Apply the Chain Rule
    The derivative of y in x is: derivative of the outer function computed in g(x), that is 1 over g(x), for the derivative of the inner function in x, i.e. g'(x)


    y' = {\frac{1}{g(x)}}\cdot g'(x)

    Substituting g(x) = \ln(x) e g'(x) = \frac{1}{x} in the formula we find

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

    This is the final result.