Problem 8 - Calculus Early Transcendentals

Question

Find the derivative of the following functions. $$f(t)=t^{11}$$

Step-by-Step Solution

Answer

Answer: The derivative of the function $$f(t)=t^{11}$$ is $$11t^{10}$$.

Step 1: Identify the exponent in the given function

In the given function, the exponent is 11. So, our n value is 11.

Step 2: Apply the power rule

Now, using the power rule, we'll find the derivative of the given function: $$\frac{d}{dt}t^{11} = 11 \cdot t^{11-1}$$

Step 3: Simplify the expression

Simplify the expression to get the final derivative: $$\frac{d}{dt}t^{11} = 11 \cdot t^{10}$$ So, the derivative of the function $$f(t)=t^{11}$$ is $$11t^{10}$$.

Key Concepts

Power Rule

The Power Rule is a fundamental theorem in calculus that greatly simplifies the process of differentiation, especially when dealing with polynomial functions. When you have a function in the form of \( f(x) = x^n \), where \( n \) is a constant exponent, the Power Rule enables you to easily find its derivative. To apply the Power Rule, follow this simple formula:

• Identify the exponent \( n \) in your function.
• Multiply the entire term by this exponent.
• Subtract one from the exponent to find the new exponent.
• Your derivative will then be \( nx^{n-1} \).

For example, if you have \( f(t) = t^{11} \), by applying the Power Rule, the derivative becomes \( 11t^{10} \). This process turns a complex-looking term into a manageable expression that gives us the slope at any point \( t \) along the function curve.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. In polynomial functions like \( f(t) = t^{11} \), exponentiation describes raising the base \( t \) to the power of \( 11 \), represented as \( t^{11} \). Here’s what each part means:

• The base \( t \) represents the number that is multiplied by itself.
• The exponent 11 signifies how many times the base is used as a factor in multiplication.

As a simple example, \( t^3 = t \times t \times t \). In our original exercise, \( t^{11} \) means multiplying \( t \) by itself 11 times. This concept is crucial for employing the Power Rule, as the derivative heavily depends on manipulating this exponent. Thus, understanding how exponentiation works allows you to apply the Power Rule efficiently in calculus.

Differentiation

Differentiation is a key concept in calculus, primarily used to find the derivative of a function. A derivative represents the rate at which a function is changing at any given point and is fundamental in understanding the behavior of functions. When differentiating a function like \( f(t) = t^{11} \), the goal is to determine \( \frac{d}{dt}t^{11} \). This process involves applying established rules such as the Power Rule to simplify the function into a derivative. Differentiation has crucial applications across many fields including physics, engineering, and economics, as it provides insights into the slope of a curve, optimization problems, and rates of change. By mastering the technique of differentiation, you can solve complex problems and better understand how different mathematical functions behave and transform.