Problem 60 - Thomas Calculus in SI Units

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

The given function is\[ y = t^{1-e} \]where \( t \) is the independent variable. We need to find the derivative of \( y \) with respect to \( t \).

Step 2: Apply the Power Rule

The power rule for differentiation states that if \( y = t^n \), then the derivative \( \frac{dy}{dt} = n \cdot t^{n-1} \). Here, our exponent \( n = 1-e \).

Step 3: Differentiate Using the Power Rule

Applying the power rule to our function:\[ \frac{dy}{dt} = (1-e) \cdot t^{(1-e)-1} \]This simplifies to:\[ \frac{dy}{dt} = (1-e) \cdot t^{-e} \]

Step 4: Finalize the Derivative

We derived that the derivative of the function is:\[ \frac{dy}{dt} = (1-e) \cdot t^{-e} \] This expression gives the rate of change of \( y \) with respect to \( t \).

Key Concepts

Derivative Calculation

In mathematics, the derivative provides a way to show how a function changes as its input changes. More specifically, the derivative represents the rate at which a function is changing at any given point and is a fundamental tool in calculus.
For the function in the exercise, which is represented as \( y = t^{1-e} \), the task is to find the derivative with respect to the variable \( t \).

• The derivative tells us how the dependent variable \( y \) changes when the independent variable \( t \) changes.
• To compute the derivative, different rules and formulas are applied depending on the form of the function.

In this exercise, we utilize the power rule because the function is in the form of a power of \( t \). Understanding how to compute derivatives is crucial in analyzing the behavior of functions, especially in physics, engineering, and economics where rates of change are essential.

Power Rule of Differentiation

The power rule of differentiation is a straightforward method used to find the derivative of functions that are powers of a variable. When a function is expressed as \( y = t^n \), the power rule indicates its derivative is \( \frac{dy}{dt} = n \cdot t^{n-1} \).

• The exponent \( n \) becomes the coefficient of the derivative, multiplying the original power of the variable, and then we reduce the power by one.
• This rule simplifies the process of differentiation, eliminating complex algebraic manipulations needed otherwise.

For the given function \( y = t^{1-e} \), applying the power rule is effective. Here, the exponent \( 1-e \) is treated as \( n \). Applying the rule, we derive that \( \frac{dy}{dt} = (1-e) \cdot t^{-e} \). This form is achieved after reducing the power of \( t \) by one and multiplying by the original exponent.
The power rule is crucial because it gives us a quick method to find derivatives, which can further be used to understand motion, growth, and decay in various contexts.

Rate of Change

Understanding the concept of rate of change is pivotal in many scientific fields. The rate of change is essentially the derivative, which indicates how a quantity changes in relation to another.
For the function \( y = t^{1-e} \), the derivative \( \frac{dy}{dt} = (1-e) \cdot t^{-e} \) effectively expresses how \( y \) changes as \( t \) varies.

• The expression \( (1-e) \cdot t^{-e} \) shows the rate of change depending on the value of \( e \).
• If the rate of change is positive, \( y \) increases as \( t \) increases. If it is negative, \( y \) decreases as \( t \) increases.

Rate of change is fundamental in understanding behaviors such as velocity (how position changes over time), acceleration (how velocity changes over time), and much more.
By calculating derivatives and understanding rate of change, one gains insight into the dynamics and variations within systems, facilitating predictions and adjustments based on different conditions.