Question

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

Answer 1

Use the power rule for differentiation:

`\frac{\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))^{(1)/(4)}`

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

`\frac{\text{d}}{\text{d}x} (\ln(x))^{(1)/(4)} = (1)/(4)(\ln(x))^{(1)/(4)-1}(1)/(x)`

Which you can rewrite as

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