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How do you find the derivative of # (ln(x^2))/x^2#?

Answer 1

Isaiah Allen

#2 * frac{1 - ln x^2}{x^3}#

#D_x ln(x^2)/x^2 = frac{x^2 D_x ln x^2 - ln x^2 D_x x^2}{x^4}#

# = frac{x^2 1/x^2 D_x x^2 - 2x ln x^2}{x^4}#

# = frac{2x - 2x ln x^2}{x^4}#

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Answer 2

Paisley Johnson

To find the derivative of (\frac{{\ln(x^2)}}{{x^2}}), you can use the quotient rule. The quotient rule states that if you have a function (f(x)) divided by (g(x)), the derivative is given by (\frac{{f'(x)g(x) - f(x)g'(x)}}{{[g(x)]^2}}). Applying this rule to the given function, the derivative is:

(\frac{{\ln(x^2)}{x^2} - \frac{{2x\cdot x^2}}{{x^2}}}{(x^2)^2})

(= \frac{{2x \cdot \ln(x^2) - 2x}}{{x^4}})

(= \frac{{2x[\ln(x^2) - 1]}}{{x^4}})

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.