Problem 10 - University Calculus: Early Transcendentals

Question

Find the indicated derivatives. $$\frac{d v}{d t}\( if \)v=t-\frac{1}{t}$$

Step-by-Step Solution

Answer

The derivative is \( 1 - \frac{1}{t^2} \).

Step 1: Identify the Function to Differentiate

We are given the function \( v(t) = t - \frac{1}{t} \). We need to find the derivative \( \frac{d v}{d t} \).

Step 2: Differentiate Each Term Individually

Differentiate the term \( t \) with respect to \( t \). This is a simple power rule operation, where the derivative of \( t \) is 1. For the second term \( -\frac{1}{t} \), rewrite it as \( -t^{-1} \) before differentiating.

Step 3: Apply the Power Rule to the Second Term

Differentiate \( -t^{-1} \) using the power rule. The derivative of \( t^n \) is \( n\cdot t^{n-1} \). Thus, for \( -t^{-1} \), the derivative is \( -1\cdot t^{-2} = -\frac{1}{t^2} \).

Step 4: Combine the Results

Combine the derivatives of each term. The derivative of \( v(t) = t - \frac{1}{t} \) is \( \frac{d v}{d t} = 1 - \frac{1}{t^2} \).

Step 5: Write the Final Answer

The final expression for the derivative is \( \frac{d v}{d t} = 1 - \frac{1}{t^2} \). You can now use this to find the rate of change of \( v \) with respect to \( t \).

Key Concepts

Understanding the Power Rule

The power rule is a fundamental concept in calculus used to find the derivative of functions that are expressed as powers of a variable. When you have a function in the form of \( t^n \), the power rule states that the derivative will be \( n \cdot t^{n-1} \). This means you multiply the coefficient (which is \( 1 \) if not explicitly written) by the exponent and subtract one from the exponent to find the derivative.

Here’s a simple breakdown:

• For \( t^2 \), the derivative is \( 2 \cdot t^1 = 2t \).
• For \( t^3 \), it becomes \( 3 \cdot t^2 = 3t^2 \).

The power rule is exceptionally useful because it allows you to quickly differentiate a wide range of polynomial expressions. In our exercise, we applied the power rule to the term \( -t^{-1} \), which became \( -1 \cdot t^{-2} = -\frac{1}{t^2} \). This step simplifies the process of finding derivatives and is essential for handling algebraic expressions in calculus.

Mastering Derivative Calculation

Derivative calculation is the process of finding the derivative, or the rate of change, of a function with respect to a variable. In calculus, the derivative provides us with the slope of the tangent line to the function at any given point on its curve. This is crucial for understanding how a function behaves and varies over its domain.

Here’s how we approached the problem:

• First, identify the function to differentiate, in this case, \( v(t) = t - \frac{1}{t} \).
• Differentiate each term separately by using calculus rules like the power rule.

In derivative calculation, you often use known rules and operations such as the power rule, product rule, or chain rule. This process allows you to determine the function’s instantaneous rate of change. By applying these rules to each term in the function, you systematically find the derivative, as we did with \( v(t) \) to get \( \frac{d v}{d t} = 1 - \frac{1}{t^2} \).

Getting to Know a Function of a Variable

A function of a variable is an expression where you can substitute different values for the variable to get various outputs. Think of it like a machine that processes input values and produces corresponding results. Here, our function involves the variable \( t \), written as \( v(t) = t - \frac{1}{t} \). This setup shows us how the function value \( v \) changes as \( t \) changes.

Understanding this concept is fundamental in calculus because it gives insight into how variables interact in an equation or real-world situation. For instance, as \( t \) becomes very large or very small, \( v(t) \) will exhibit different behaviors:

• When \( t \) is large, \( \frac{1}{t} \) gets smaller, so \( v(t) \approx t \).
• When \( t \) is small, \( \frac{1}{t} \) becomes significant, affecting \( v(t) \) more.

Functions of a variable help you model and predict behaviors and are central to understanding dynamic systems in mathematics and sciences.