Problem 53 - Calculus of a Single Variable

Question

In Exercises 41–64, find the derivative of the function. $$ g(t)=\frac{\ln t}{t^{2}} $$

Step-by-Step Solution

Answer

The derivative of the function \( g(t) = \frac{\ln t}{t^{2}} \) is \( g'(t) = \frac{1 - 2t\ln{t}}{t^{3}} \).

Step 1: Identify the Components of the Quotient

First identify function f(t), the numerator, as \( \ln t \) and g(t), the denominator, as \( t^{2} \).

Step 2: Compute the Derivatives f'(t) and g'(t)

The derivative f'(t) of the function f(t) = \( \ln t \) is \( \frac{1}{t} \) and the derivative g'(t) of the function g(t) = \( t^{2} \) is \( 2t \). So we have f'(t) = \( \frac{1}{t} \) and g'(t) = \( 2t \).

Step 3: Apply the Quotient Rule

Now plug these functions and their derivatives into the quotient rule formula: \[ g'(t) = \frac{f'(t)g(t) - f(t)g'(t)}{[g(t)]^2} \] This gives: \[ g'(t) = \frac{ [\frac{1}{t}] [t^{2}] - [\ln t] [2t]}{[t^{2}]^{2}} \]

Step 4: Simplify Result

Simplify above result to: \[ g'(t) = \frac{1 - 2t\ln{t}}{t^{3}} \]

Key Concepts

Quotient Rule

Understanding the quotient rule is essential when dealing with the derivative of a function that is the ratio of two other functions. The quotient rule tells us that for two differentiable functions, say, u(t) and v(t), the derivative of their quotient u(t)/v(t) is given by:
\[ \frac{d}{dt}\left(\frac{u(t)}{v(t)}\right) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^{2}} \]
This formula seems complex at first glance, but breaking it down helps in understanding each part. The numerator of this derivative expression is the difference between the product of the derivative of the numerator and the denominator and the product of the numerator and the derivative of the denominator. The denominator of the derivative expression is simply the square of the original denominator.

Differentiation

Differentiation is a fundamental part of calculus, which focuses on finding how a function changes at any given point. It describes the rate at which the function's value changes with respect to its input, called the derivative. For example, in the function f(t) = t^2, differentiating with respect to t gives us f'(t) = 2t, which means for each unit increase in t, the function's value increases by an amount proportional to 2t. When it comes to more complex functions, such as those involving logarithms or ratios, specific rules like the quotient rule discussed earlier come into play to streamline the process.

Natural Logarithm

The natural logarithm, denoted as ln(t), is a logarithm to the base e, where e is an irrational and transcendental number approximately equal to 2.71828. The natural logarithm of t asks the question: to what power must we raise e to obtain the number t? In calculus, the derivative of the natural logarithm function is unique. Taking the derivative of ln(t) yields 1/t. This derivative is exceptionally useful and appears often in problems involving growth and decay, where processes occur continuously. Moreover, due to its continuous nature, the natural logarithm facilitates the solving of differential equations and is pivotal in understanding change over an infinitesimal interval.