Problem 15 - Calculus Early Transcendentals

Question

Find the derivative of the following functions. See Example 4 of Section 3.1 for the derivative of \(\sqrt{x}\). $$p(x)=8 x$$

Step-by-Step Solution

Answer

Answer: p'(x) = 8

Step 1: Identify the power of x

In this case, the power of x is n = 1.

Step 2: Apply the power rule

Using the power rule, we can find the derivative of the function p(x): $$p'(x) = n \cdot a \cdot x^{n-1} = 1 \cdot 8 \cdot x^{1-1}$$

Step 3: Simplify the expression

Simplify the expression for p'(x): $$p'(x) = 8 \cdot x^0 = 8 \cdot 1 = 8$$ The derivative of the function p(x) = 8x is: $$p'(x) = 8$$

Key Concepts

Power Rule

When taking derivatives in calculus, the power rule is an essential technique that simplifies the process significantly. This rule applies when you are finding the derivative of a term with a variable raised to a power. Essentially, it states that if you have a function of the form f(x) = ax^n, the derivative f'(x) is found by multiplying the power n by the coefficient a, and then subtracting one from the power.

For instance, in the function p(x) = 8x, to apply the power rule, you identify the exponent of x, which is 1 (since x^1 = x). The derivative, denoted as p'(x), is then computed as 1 * 8 * x^(1-1), which simplifies to 8 * x^0 = 8. Notice that x^0 is always 1, so it leaves us with just the coefficient, 8.

Understanding the power rule equips you to quickly find derivatives without the need for elaborate calculations, making it a cornerstone in the study of calculus.

Derivative of Polynomial Functions

Polynomial functions are expressions that involve sums of powers of a variable, typically written in the form p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where a_n are coefficients, and n is a non-negative integer. Taking derivatives of polynomial functions relies heavily on the power rule, as each term of the polynomial can be treated separately.

To find the derivative of a polynomial function, you need to apply the power rule to each term involving a variable. For example, in our exercise, the function p(x) = 8x is a linear polynomial where the derivative is simply 8, as calculated in the exercise. This process is the same for polynomials of higher degrees; you would apply the power rule to each term and sum the results to find the derivative of the entire function.

It's important to remember that the derivative of a constant term (like a_0) is zero since it doesn't contain a variable, and hence its rate of change with respect to x is zero. Knowing this, we can efficiently differentiate more complex polynomial functions.

Calculus Basics

Calculus is a branch of mathematics that deals with change and motion. At the heart of calculus are derivatives and integrals. The derivative is a measure of how a function changes as its input changes. In other words, it tells us the rate at which values are changing. In the realm of real-world applications, derivatives are used to calculate rates like speed, acceleration, and the slope of a curve at a particular point.

Understanding the basic principles of calculus, such as limits, continuity, and the concept of differentiability are fundamental. For beginners, mastering the straightforward rules like the power rule for derivatives is a good foundation, as it allows to quickly move onto more complex problems without getting bogged down in intricate details.

The ability to comprehend and execute derivative calculations paves the way for understanding more advanced concepts in calculus, such as optimization, where one might seek to find maximum and minimum values of functions, which has practical implications in diverse fields such as engineering, economics, and physics.