Problem 19 - Calculus Early Transcendentals

Question

Find \(d x / d t\) $$ x=t^{2}-t $$

Step-by-Step Solution

Answer

\( \frac{dx}{dt} = 2t - 1 \)

Step 1: Identify the Function

The given function to differentiate is \( x = t^2 - t \). This is a polynomial function where \( x \) is expressed in terms of \( t \).

Step 2: Apply the Power Rule

To differentiate \( x = t^2 - t \) with respect to \( t \), use the power rule. The power rule states that the derivative of \( t^n \) is \( n \cdot t^{n-1} \).

Step 3: Differentiate Each Term

Differentiate the term \( t^2 \) using the power rule: \( \frac{d}{dt} (t^2) = 2t \). Then differentiate the term \( -t \): \( \frac{d}{dt} (-t) = -1 \).

Step 4: Combine the Derivatives

Combine the derivatives of each term: \( \frac{dx}{dt} = 2t - 1 \). This is the derivative of \( x \) with respect to \( t \).

Key Concepts

Polynomial Function

A polynomial function is an expression that involves variables raised to whole-number exponents and their coefficients. In the given exercise, we have a polynomial function, specifically, \( x = t^2 - t \). A polynomial is composed of terms such as constants, variables raised to powers, and their products. Each term in the polynomial is a simple mathematical operation.

Key features of polynomial functions include:

• They can contain multiple terms.
• Exponents of variables are non-negative integers.
• The coefficients are real numbers.

Recognizing polynomial functions is important because it determines which differentiation rules to use. Polynomial functions are smooth and continuous, making them easy to manipulate mathematically.

Power Rule

The power rule is a fundamental differentiation rule used when dealing with polynomials. It simplifies finding the derivative of each term in a polynomial separately. The rule states that if you have \( f(t) = t^n \), then the derivative, denoted by \( f'(t) \), is \( n \cdot t^{n-1} \).

To apply the power rule correctly, follow these steps:

• Identify the exponent \( n \) of the term you are differentiating.
• Multiply the entire term by \( n \).
• Subtract one from the exponent \( n \).

This method is both efficient and straightforward, making it ideal for differentiating polynomial terms as in our example where we needed to find \( \frac{d}{dt}(t^2) \) which resulted in \( 2t \).

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function’s value with respect to a variable. In the context of our example, we differentiated \( x = t^2 - t \) with respect to \( t \) to find \( \frac{dx}{dt} \).

Why differentiate? Here are a few reasons:

• Understand how a function changes at any point.
• Determine the slope of the tangent at any point on a curve represented by the function.
• Find local maxima and minima of functions.

In our exercise, differentiation was carried out term-by-term: \( \frac{d}{dt}(t^2) = 2t \) and \( \frac{d}{dt}(-t) = -1 \). The final derivative is a combination of these individual derivatives: \( \frac{dx}{dt} = 2t - 1 \). This shows the rate of change of \( x \) with respect to \( t \).