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Chapter 5: Problem 53

Find the derivative of the function. $$ g(t)=\frac{\ln t}{t^{2}} $$

Short Answer

Expert verified
The derivative of the function \(g(t) = \frac{\ln t}{t^2}\) is \(g'(t) = \frac {t - 2t \ln t}{t^4}\)

Step by step solution

01

Identify \(u\) and \(v\)

Let \(u = \ln t\) and \(v = t^2\). So the given function can be rewritten as \(g(t) = \frac{u}{v}\).
02

Compute derivatives of \(u\) and \(v\)

The derivative of \(u\) with respect to \(t\) is \(u' = \frac{1}{t}\). The derivative of \(v\) with respect to \(t\) is \(v' = 2t\).
03

Apply the quotient rule

According to the quotient rule, \(g'(t) = \frac {v(u') - u(v')}{v^2}\). After substituting all known values, we get \(g'(t) = \frac {t^2(\frac{1}{t}) - \ln t(2t)}{(t^2)^2} = \frac {t - 2t \ln t}{t^4}\) after simplifying.

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