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Lemur [1.5K]
2 years ago
9

{ lnx } " align="absmiddle" class="latex-formula">
how do I differentiate this
Mathematics
1 answer:
Kamila [148]2 years ago
3 0

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}}

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