Problem 72 - University Calculus: Early Transcendentals

Question

Find the derivative of \(y\) with respect to the given independent variable. $$y=t^{1-e}$$

Step-by-Step Solution

Answer

The derivative of \( y = t^{1-e} \) with respect to \( t \) is \((1-e) t^{-e}\).

Step 1: Identify the function

The given function is \( y = t^{1-e} \), where \( t \) is the independent variable and \( e \) is the base of the natural logarithm, approximately equal to 2.71828.

Step 2: Recognize the Power Rule

To find the derivative of a function in the form \( t^n \), we use the power rule, which states that \( \frac{d}{dt} t^n = n \cdot t^{n-1} \).

Step 3: Apply the Power Rule

Using the power rule, the derivative of \( y = t^{1-e} \) with respect to \( t \) is \( \frac{d}{dt} y = (1-e) \cdot t^{(1-e)-1} \).

Step 4: Simplify the Expression

Simplifying the derivative expression, we get \( \frac{d}{dt} y = (1-e) \cdot t^{-e} \).

Key Concepts

Power Rule

The power rule is a fundamental concept used in calculus to find the derivative of a function that is expressed as a power of a variable. In simple terms, if you have a function like \( t^n \), where \( n \) is any real number, the power rule allows us to differentiate it by following these steps:

• Bring down the exponent \( n \) as a coefficient in front of the variable \( t \).
• Subtract one from the exponent \( n \).

This formula can be written as \( \frac{d}{dt} t^n = n \cdot t^{n-1} \).
For the exercise you are looking at, we apply this rule to the function \( y = t^{1-e} \). Here, the exponent \( 1-e \) takes the place of \( n \). By applying the power rule, we obtain the derivative: \( (1-e) \cdot t^{(1-e)-1} \), which simplifies to \( (1-e) \cdot t^{-e} \). This step-by-step approach of the power rule makes finding derivatives of simple power functions quick and efficient.

Independent Variable

In mathematical functions, the independent variable is the one whose variation does not depend on other variables in the equation. It is the input of a function, and in calculus, it's important to know which variable is independent when differentiating. Think of it as the variable you control or the one that stands alone.
For example, in the function \( y = t^{1-e} \), \( t \) is the independent variable. As such, calculations, derivatives, and interpretations depend on how \( t \) changes.
The independent variable sets the stage for applying rules like the power rule in differentiation. It gives us a clear direction in terms of which variable is responsible for the function's output changes. Understanding which variable is the independent one ensures the correct application of differentiation rules.

Natural Logarithm

The natural logarithm, denoted as \( \, \ln \, \) or sometimes as \( \, \log_e \, \), is a special kind of logarithm where the base is the constant \( e \), approximately equal to 2.71828. This constant \( e \) is irrational, much like \( \pi \), and it arises naturally in many areas of mathematics, especially in calculus.

• \( e \) is particularly important in continuous growth and compound interest calculations.
• The derivative of \(e^x\) with respect to \(x\) is \(e^x\), showing its unique property.

In the original exercise, \( e \) appears in the exponent of \( t \) in the expression \( t^{1-e} \), which highlights how pervasive \( e \) is in calculations. The presence of \( e \) often indicates a connection to exponential growth or decay, forms of growth processes or natural phenomena that happen consistently over time. Recognizing \( e \) and its related functions, like the natural logarithm, is crucial for understanding continuous change in calculus.