Problem 20 - University Calculus: Early Transcendentals

Question

Find \(d s / d t\). $$s=t^{2}-\sec t+5 e^{t}$$

Step-by-Step Solution

Answer

The derivative is \( \frac{ds}{dt} = 2t - \sec t \tan t + 5e^t \).

Step 1: Recognize the Derivative Notation

The problem asks us to find the derivative of the function with respect to time \( t \), i.e., \( \frac{ds}{dt} \). This means we need to differentiate each term of the function \( s(t) = t^2 - \sec t + 5e^t \).

Step 2: Differentiate the Polynomial Term

The first term \( t^2 \) is a polynomial. Using the power rule \( \frac{d}{dt}[t^n] = nt^{n-1} \), the derivative of \( t^2 \) is \( 2t \).

Step 3: Differentiate the Trigonometric Term

The second term is \(-\sec t\). Recall that the derivative of \( \sec t \) is \( \sec t \tan t \). Therefore, the derivative for \(-\sec t\) is \(-\sec t \tan t\).

Step 4: Differentiate the Exponential Term

The third term is \( 5e^t \). The derivative of \( e^t \) is \( e^t \), so the derivative of \( 5e^t \) is \( 5e^t \), as constants multiply through differentiation.

Step 5: Combine the Derivatives

Now, sum up all the derivatives from each term: \[ \frac{ds}{dt} = 2t - \sec t \tan t + 5e^t \].

Step 6: Simplify and State the Final Expression

The final expression for the derivative of \( s \) with respect to \( t \) is \( \frac{ds}{dt} = 2t - \sec t \tan t + 5e^t \). There is nothing further to simplify, so this is our answer.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change. For a function of a single variable, the derivative at a particular point informs us how the function changes as the variable changes. In the given exercise, differentiation is employed to determine how the function \( s(t) = t^2 - \sec t + 5e^t \) changes with respect to time \( t \). This process involves taking the derivative of each term of the function and combining them. By following systematic rules for differentiation, we can simplify complex functions to understand their behavior over time. This is especially useful in physics and engineering where change and rates of change are crucial.

Trigonometric Derivatives

Trigonometric functions like \( \sin t \), \( \cos t \), and \( \sec t \) have specific rules for differentiation. In this exercise, the trigonometric term \( -\sec t \) needs to be differentiated. Each trigonometric function has a known derivative:

• The derivative of \( \sin t \) is \( \cos t \).
• The derivative of \( \cos t \) is \( -\sin t \).
• The derivative of \( \sec t \) is \( \sec t \tan t \).

When solving \( \frac{ds}{dt} \), \(-\sec t\) becomes \(-\sec t \tan t\). It's crucial to remember these rules and apply them correctly. Doing so helps simplify the process and ensures accuracy in finding how trigonometric components of a function change.

Polynomial Derivatives

Polynomial derivatives are calculated by applying the power rule. This rule states that if you have a term like \( t^n \), its derivative is \( nt^{n-1} \). In our function \( s(t) \), one term is \( t^2 \). Using the power rule,

• The derivative of \( t^2 \) is \( 2t \).

Polynomial functions usually present as manageable terms when differentiating, as each term's power is reduced by one, making the calculation straightforward. Differentiation of polynomials showcases how they change gradually and linearly, essential when examining functions in calculus.

Exponential Derivatives

Exponential functions, such as \( e^t \), have a unique property when it comes to differentiation. The derivative of \( e^t \) is particularly simple: it is itself, \( e^t \). This differentiates it from other types of functions, where the form changes upon taking a derivative. When a coefficient multiplies an exponential term, like \( 5e^t \), the coefficient remains unchanged. The derivative of \( 5e^t \) is

• \( 5e^t \)

This behavior makes exponential functions easy to work with in calculus. Exponential derivatives are pivotal in modeling growth and decay processes, be it population growth or radioactive decay, emphasizing their real-world applications.