Problem 13 - Applied Calculus

Question

Find the derivative. Assume \(a, b, c, k\) are constants. $$y=3 x^{2}+7 x-9$$

Step-by-Step Solution

Answer

The derivative of the function is \( 6x + 7 \).

Step 1: Identify the Function Type

The given function is a polynomial function expressed as \( y = 3x^2 + 7x - 9 \). Each term involves a power of \( x \) with a numerical coefficient.

Step 2: Apply the Power Rule for Derivatives

To differentiate the function, apply the power rule \( \frac{d}{dx}x^n = nx^{n-1} \) to each term separately.

Step 3: Differentiate 3x²

Using the power rule, the derivative of \( 3x^2 \) is calculated as follows: \( \frac{d}{dx}(3x^2) = 6x \).

Step 4: Differentiate 7x

The derivative of \( 7x \) is simply the constant itself, \( \frac{d}{dx}(7x) = 7 \), because the power of \( x \) is 1.

Step 5: Differentiate the Constant Term -9

The derivative of a constant term, such as \(-9\), is \(0\) because the rate of change of a constant is always zero.

Step 6: Combine the Derivatives

Combine the derivatives of all terms to find the derivative of the function: \( \frac{dy}{dx} = 6x + 7 + 0 \).

Step 7: Simplify the Expression

Simplifying the expression from the previous step gives us \( \frac{dy}{dx} = 6x + 7 \).

Key Concepts

Power Rule

Understanding the power rule is fundamental when you begin to learn about derivatives in calculus. The power rule states that to differentiate a term of the form \( x^n \), you multiply the exponent \( n \) by the coefficient of \( x \) and then decrease the exponent by one. This simple rule, written as \( \frac{d}{dx}x^n = nx^{n-1} \), allows us to quickly determine the derivative of a polynomial term.

• Apply the exponent as a multiplier to the coefficient.
• Subtract one from the power of \( x \).
• This gives the derivative of the term.

For example, using the power rule on the term \( 3x^2 \) involves multiplying the coefficient (3) by the exponent of \( x \) (2) to get 6, and then reducing the exponent of \( x \) by one to get \( 6x^1 = 6x \). This is why the derivative of \( 3x^2 \) is \( 6x \).
Understanding how to apply the power rule to each separate term is key when differentiating more complex polynomial functions.

Polynomial Function

Polynomial functions are a type of function composed of variables raised to whole number powers and multiplied by coefficients. They are written in the form \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( a_n, a_{n-1}, \ldots, a_0 \) are constants, and \( n \) is a non-negative integer.
These functions are very versatile:

• They can represent a wide variety of curves.
• The degree of the polynomial (the highest power of \( x \) in the expression) determines a lot about the shape and behavior of the graph.
• Simple operations can be performed on them, including addition, subtraction, multiplication, and, importantly for calculus, differentiation.

In the given exercise, the function is \( y = 3x^2 + 7x - 9 \), a polynomial of degree 2 known as a quadratic function. It's made up of three distinct terms, each of which is easily differentiated using the power rule.
Polynomial functions such as these are foundational in calculus and many areas of mathematics because they model numerous real-world processes.

Differentiation Steps

Differentiation is the process of finding the derivative, which represents the rate of change of a function with respect to a variable. In our exercise, differentiating the polynomial function \( y = 3x^2 + 7x - 9 \) demonstrates a straightforward method.Here is the step-by-step differentiation process applied:

• **Identify:** Recognize the function type, which is a polynomial.
• **Apply the Power Rule:** Use the power rule to find the derivative of each term.
• **3x²:** Differentiate \( 3x^2 \) to get \( 6x \).
• **7x:** The term \( 7x \) differentiates to 7 because \( x^1 \) reduced by one gives \( x^0 = 1 \).
• **-9:** The constant differentiates to zero.
• **Combine:** Add all the derived terms together: \( 6x + 7 + 0 \).
• **Simplify:** The expression is simplified to \( \frac{dy}{dx} = 6x + 7 \).

Each of these steps logically builds on the others, making differentiation of polynomial functions a structured and reliable process. Differentiation is crucial in calculus as it helps understand how variables change relative to each other, aiding in solving real-life problems.