Problem 64 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The derivative is \( \frac{dy}{dt} = (1-e) \cdot t^{-e} \).

Step 1: Identify the Function

The function given is \( y = t^{1-e} \) where \( e \) is a constant, specifically the base of the natural logarithm. We need to find the derivative of \( y \) with respect to \( t \).

Step 2: Apply the Power Rule

The power rule for differentiation is \( \frac{d}{dt}[t^n] = n \cdot t^{n-1} \). Identify \( n = 1-e \) in the equation \( y = t^{1-e} \).

Step 3: Differentiate the Function

Using the power rule, differentiate: \( \frac{d}{dt}[t^{1-e}] = (1-e) \cdot t^{(1-e)-1} = (1-e) \cdot t^{-e} \).

Step 4: Express the Result

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

Key Concepts

Power Rule

The power rule is a fundamental concept in calculus that greatly simplifies the process of finding the derivative of any power of a variable. If you have a function with a variable raised to a power, like \( f(t) = t^n \), the power rule states that the derivative of this function with respect to the variable is given by \[ \frac{d}{dt}[t^n] = n imes t^{n-1} \]This means you bring down the exponent as a coefficient in front of the variable and then reduce the exponent by one.
The power rule only works when the exponent is a constant number. This makes it a straightforward tool to differentiate algebraic functions.
In our exercise, we have a function \( y = t^{1-e} \) where \( 1-e \) is the exponent. By applying the power rule, the differentiation becomes manageable, turning the original expression into a simpler form where differentiation is carried out.

Differentiation

Differentiation is one of the core operations in calculus, closely tied to the concept of deriving rates of change. It involves finding the derivative, which tells us how a function changes as its input changes.
The derivative can be thought of as the slope of the tangent line to the graph of the function at any point, representing the rate of change.
In the given exercise, we differentiate the function \( y = t^{1-e} \) with respect to \( t \) to find how \( y \) changes as \( t \) changes. For functions of the form \( t^n \), the power rule allows for fast and accurate differentiation.

• The first step is identifying the components of your function, such as the base and the exponent.
• Next, apply any necessary rules, like the power rule, to either reduce the complexity or focus directly on differentiability.
• Finally, perform the actual mathematical operations that yield the derivative.

Through differentiation, we gain insight into the behavior of a function, which is why it's often used to study dynamic systems in physics, economics, and other fields.

Natural Logarithm Constant

The number \( e \) is a unique constant approximately equal to 2.71828 and known as the base of the natural logarithm. It arises in many areas of mathematics, particularly where growth processes are modeled.
The function \( e^x \), which describes exponential growth or decay, is significant because its derivative—expressing its rate of growth—is also \( e^x \). This property makes \( e \) central to calculus and natural systems.
In our context, \( e \) appears as part of the exponent in the derivative. Understanding \( e \) is crucial because it characterizes a natural mathematical process.

• \( e \) occurs naturally in processes that exhibit continuous growth, like compound interest or population growth.
• It's involved in complex computations, including limits and asymptotic behavior in analysis.
• In the given equation \( y = t^{1-e} \), \( e \) contributes to defining how the function behaves in relation to the variable \( t \).

Knowing about the natural logarithm constant helps in grasping the deeper calculus concepts and the natural world interpretations linked to exponential growth and decay.