Problem 12 - Calculus Early Transcendentals

Question

Find the derivative of the following functions. $$f(v)=v^{100}$$

Step-by-Step Solution

Answer

Answer: The derivative of the function $$f(v) = v^{100}$$ is $$f'(v) = 100v^{99}$$.

Step 1: Identify the function and its exponent

We have the function $$f(v) = v^{100}$$. Here, the variable is $$v$$, and the exponent is $$100$$.

Step 2: Apply the power rule

Using the power rule for derivatives, we get $$f'(v) = 100v^{100-1}$$

Step 3: Simplify the derivative

Now, simplify the exponent: $$f'(v) = 100v^{99}$$ That's it! The derivative of the function $$f(v) = v^{100}$$ is $$f'(v) = 100v^{99}$$.

Key Concepts

Power Rule for Differentiation

When studying calculus, one of the essential tools at a student's disposal is the power rule for differentiation. This rule is a shortcut that makes finding the rate of change of a function, raised to a power, a breeze. Let's focus on this rule which states that if you have a function of the form \( f(x) = x^n \), where \( n \) is any real number, the derivative of that function is given by \( f'(x) = nx^{n-1} \).

Applying this rule is straightforward: Start by bringing down the exponent as a coefficient in front of the variable, and then subtract one from the original exponent to get the new exponent in the derivative. For instance, if you had the function \( f(v) = v^{100} \), the derivative would be \( f'(v) = 100v^{99} \), as seen in our original exercise. It's crucial to remember that this rule only applies when the base of the power is the variable you are differentiating with respect to.

Simplifying Exponents

Exponents can seem daunting at first, but understanding how to simplify them is critical for mastering algebra and calculus. When we simplify exponents, we follow a set of rules to write expressions in a more manageable form. In the case of our differentiation problem, simplifying simply meant applying the power rule and reducing the exponent of \( v \) from 100 to 99. However, there are other situations where one might need to multiply exponents, divide them, or apply them to products and quotients.

To illustrate, if you had an expression like \( x^m \times x^n \), you would add the exponents to simplify it to \( x^{m+n} \). In contrast, if you were dividing, such as in \( x^m / x^n \), you would subtract the exponents, giving you \( x^{m-n} \). It's these rules that enable us to manipulate and simplify complex algebraic expressions before or after we've taken derivatives.

Calculus

Calculus is a branch of mathematics that studies how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. There are two primary branches of calculus: differential calculus concerning rates of change and slopes of curves; and integral calculus which deals with the accumulation of quantities and the areas under and between curves.

Our exercise focused on differential calculus, specifically on taking derivatives. The derivative is a fundamental concept that measures how a function changes as its input changes. It's often conceptualized as the slope of the curve at any given point and represents the rate of change, or how fast or slow something is moving. Beyond the power rule for differentiation and simplifying exponents, calculus covers a variety of techniques for dealing with more complex functions and their derivatives, including product rule, quotient rule, and chain rule, which are essential for tackling a wide range of scientific and engineering problems.