Question

$\sqrt[4]{ lnx }$
how do I differentiate this

Answer 1

Use the power rule for differentiation:

$\dfrac{\text{d}}{\text{d}x} (f(x))^k = k(f(x))^{k-1}f'(x)$

You can use this formula if you remember that a root is just a rational exponential:

$\sqrt[4]\ln(x) = (\ln(x))^{\frac{1}{4}}$

So, remembering that the derivative of the logarithm is 1/x, you have

$\dfrac{\text{d}}{\text{d}x} (\ln(x))^{\frac{1}{4}} = \frac{1}{4}(\ln(x))^{\frac{1}{4}-1}\dfrac{1}{x}$

Which you can rewrite as

$\dfrac{1}{4}(\ln(x))^{\frac{1}{4}-1}\dfrac{1}{x} =\dfrac{1}{4}(\ln(x))^{\frac{-3}{4}}\dfrac{1}{x} =\dfrac{1}{4}\dfrac{1}{\sqrt[4]{\ln(x))^3}}\dfrac{1}{x} = \dfrac{1}{4x\sqrt[4]{\ln(x))^3}}$