Problem 16 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

\(\frac{dy}{dt} = (1-e)t^{-e}\)

Step 1: Apply the Power Rule for Derivatives

To find the derivative of a function that is a power of the variable, we use the power rule. The power rule states that if \(y = t^n\), then \(\frac{dy}{dt} = nt^{n-1}\). In this case, \(y = t^{1-e}\), so \(n = 1 - e\).

Step 2: Differentiate Using the Power Rule

Apply the power rule to differentiate \(y = t^{1-e}\). The derivative is \(\frac{dy}{dt} = (1-e)t^{1-e-1}\).

Step 3: Simplify the Expression

Simplify the derivative expression. \(\frac{dy}{dt} = (1-e)t^{-e}\).

Key Concepts

Understanding Derivatives

Derivatives are a fundamental aspect of calculus. They represent the rate at which a function changes. Think of it as finding the slope of a function at any point. For example, if you have a position function, its derivative will give you the velocity function. Derivatives help us understand how a function behaves without needing to graph it. In our exercise, finding the derivative of the function allows us to determine how the function changes as the variable, in this case, "t", varies. When we say with respect to the independent variable, we are simply referring to the variable that changes freely. In most functions, this is often represented as "x" or "t" in this situation.

Exploring the Power Rule

The power rule is a handy shortcut to make finding derivatives easier. When you have a base which is a variable raised to some power, the power rule simplifies the differentiation process. Here’s how the rule works: for a function in the form of \(y = x^n\), the derivative \(\frac{dy}{dx}\) is obtained by multiplying the power \(n\) by the original function, then subtracting one from the power, hence \(nx^{n-1}\). In real-life terms, if you start with the power of the variable and bring it down as a coefficient, you then reduce the existing power by one. This rule drastically cuts down the tedious work of differentiating functions manually and is particularly useful for polynomial functions. In our example, using the power rule, we derive \(\frac{dy}{dt} = (1-e)t^{-e}\), which simplifies the differentiation task.

Concept of Independent Variable

An independent variable is the variable we change to see how it affects other variables in a function. It's like a cause-and-effect relationship: changing the independent variable changes the outcome or dependent variable. In mathematical functions, the independent variable is typically represented by "x", "t", or another letter. The function tracks what happens to the dependent variable as we manipulate the independent one. In our calculus exercise, "t" is the independent variable. It's like the control dial we can turn to analyze how our function "y" responds to different values. The derivative we found, \(\frac{dy}{dt} = (1-e)t^{-e}\), shows us how "y" changes in response to tiny changes in "t".