Problem 54 - Calculus of a Single Variable

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the functions involved

Given that \(h(t)=\frac{\ln t}{t}\), we can see that this is a quotient of two functions. We can consider \(f(t)=\ln t\) as the numerator and \(g(t)=t\) as the denominator.

Step 2: Apply the quotient rule

Given that \(h'(t)=\frac{g(t)f'(t)-f(t)g'(t)}{[g(t)]^2}\), we need to find \(f'(t)\) and \(g'(t)\). Where, \(f'(t)\) is the derivative of \(f(t) = ln(t)\) which equals \(\frac{1}{t}\) and \(g'(t)\) is the derivative of \(g(t) = t\) which equals 1.

Step 3: Substitute into quotient rule formula

Substitute \(f(t)\), \(f'(t)\), \(g(t)\) and \(g'(t)\) into the quotient rule formula. Then, \(h'(t) = \frac{t*\frac{1}{t}-\ln t*1}{(t)^2} = \frac{1-\ln t}{t^2}\)

Key Concepts

Quotient Rule

When you have a function that is the quotient of two other functions, you can use the quotient rule to find its derivative. The quotient rule helps us systematically differentiate expressions like \( h(t)=\frac{\ln t}{t} \). It can be remembered using the formula: if \( h(t) = \frac{f(t)}{g(t)} \), then the derivative \( h'(t) \) is given by:

• \[ h'(t) = \frac{g(t)f'(t) - f(t)g'(t)}{[g(t)]^2} \]

Here's a step-by-step rundown:

• Identify your numerator \( f(t) \) and your denominator \( g(t) \).


• Compute the derivative of each function. These are \( f'(t) \) and \( g'(t) \).


• Insert these derivatives into the quotient rule formula.
• Simplify your result as the final step.

Recognizing this formula and practicing its application makes differentiating quotients more straightforward.

Natural Logarithm Derivative

Understanding how to differentiate the natural logarithm function \( \ln t \) is crucial when you're working with expressions that include them. The derivative of the natural logarithm function is particularly elegant and easy to remember. If \( f(t) = \ln t \), then the derivative \( f'(t) \) is:

• \[ \frac{d}{dt}(\ln t) = \frac{1}{t} \]

This means, for every positive \( t \), the slope of the tangent to the curve at \( t \) is just \( \frac{1}{t} \). This property of the natural log makes it especially nice when simplifying problems in calculus. Remember this rule whenever a natural logarithm appears in your calculus exercises, and it will make differentiating these types of functions much simpler.

Differentiation of Functions

At the heart of calculus lies differentiation, which is determining the rate at which a function changes. When differentiating more complex functions such as \( h(t) = \frac{\ln t}{t} \), you'll combine basic differentiation rules, like power, product, and quotient rules. To effectively work through such problems, remember:

• Differentiation transforms a function into another that represents its rate of change.


• Rules like the quotient rule help to break down complex equations into manageable parts.


• Each function has specific rules, like natural logarithms, that make their derivatives straightforward.

Applying known derivatives and understanding the relationships between parts of an equation help solve problems quickly. It's essential to practice these differentiation techniques to handle any function you come across!