Finding The Derivative: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of calculus and derivatives. We'll be tackling the problem of finding the derivative of a function and evaluating it at a specific point. Specifically, we're going to figure out what f'(6) is, given that f(x) = 14x + 17. Don't worry if this sounds a bit intimidating; we'll break it down step by step and make it super easy to understand. So, grab your coffee (or your favorite beverage), and let's get started!

Understanding Derivatives: The Basics

Alright, before we jump into the nitty-gritty, let's talk about what a derivative actually is. In simple terms, the derivative of a function tells us the rate at which the function's output changes with respect to its input. Think of it like this: if you're driving a car, the function could represent your distance from home, and the derivative would represent your speed. So, the derivative gives us the instantaneous rate of change at any given point. It's essentially the slope of the tangent line to the function at that point. If you recall the fundamental theorem of calculus, you should already be familiar with derivatives! Derivatives are fundamental in calculus and have many applications in physics, engineering, economics, and other fields. Derivatives tell you the rate of change of a function at a specific point, which can be useful in several applications. In physics, derivatives are used to find the velocity and acceleration of an object. In economics, derivatives are used to find the marginal cost and revenue of a product. In engineering, derivatives are used to design and analyze systems. The derivative of a function is often denoted as f'(x), dy/dx, or df/dx. The process of finding the derivative of a function is called differentiation. Differentiation uses rules and techniques to find the derivative of a function. There are several rules for finding the derivatives of different types of functions, such as the power rule, the product rule, the quotient rule, and the chain rule. The power rule is used to find the derivative of a function that is a power of x, such as f(x) = x^n. The product rule is used to find the derivative of a function that is the product of two functions, such as f(x) = u(x)v(x). The quotient rule is used to find the derivative of a function that is the quotient of two functions, such as f(x) = u(x)/v(x). The chain rule is used to find the derivative of a composite function, such as f(x) = g(h(x)). Understanding these rules is crucial to solving more complex problems! For our example, we're dealing with a linear function, which makes things a bit easier. Let’s keep this in mind as we move forward.

Differentiating the Function: The Power Rule

Okay, now that we have a basic grasp of what a derivative is, let's find the derivative of our function, f(x) = 14x + 17. To do this, we'll use a few basic rules of differentiation. This is where it gets fun, guys!

First, let's recall the power rule. The power rule states that if we have a term like x^n, its derivative is nx^(n-1)*. For example, if f(x) = x^2, then f'(x) = 2x. We'll also use the constant rule, which states that the derivative of a constant (like 17) is always zero. The power rule is a fundamental concept in calculus. You will use it when finding derivatives of polynomials and other functions. It is important to know this rule, as it is a building block for more complex differentiation techniques, such as the product rule, the quotient rule, and the chain rule. The constant rule states that the derivative of a constant is zero. This makes sense because the derivative of a function represents its instantaneous rate of change. The rate of change of a constant is zero because the constant does not change. These rules are key to solving differentiation problems.

Now, let's apply these rules to our function. f(x) = 14x + 17. We can rewrite 14x as 14x^1. Let's break it down term by term:

• The derivative of 14x: Using the power rule, the derivative of x^1 is 1x^(1-1) which simplifies to 1x^0 which is just 1. Multiply this by 14, and we get 14.
• The derivative of 17: This is a constant, so its derivative is 0. Using the above-mentioned rules, we can find the derivative for this function.

Therefore, f'(x) = 14 + 0 = 14. The derivative of our function is a constant, 14.

Evaluating the Derivative at x = 6

Awesome, we've found the derivative, f'(x) = 14. Now, we need to find the value of the derivative at x = 6, which is f'(6). This part is incredibly simple, and you will understand why.

Since f'(x) = 14, and this is a constant, it means that the rate of change of our original function is the same no matter what the value of x is. Thus, f'(6) = 14. It's that easy, guys! In this case, there's no x to substitute with the value of 6. Because the derivative is a constant, its value doesn't change with x. This property simplifies the process and allows us to quickly find the solution.

Summary and Key Takeaways

Alright, let's quickly recap what we've done:

1. Understanding Derivatives: We talked about what derivatives represent – the instantaneous rate of change of a function.
2. Differentiating the Function: We used the power rule and the constant rule to find the derivative of f(x) = 14x + 17, which turned out to be f'(x) = 14.
3. Evaluating at x = 6: We found that f'(6) = 14, as the derivative is a constant.

Key takeaways:

• The derivative of a linear function (like ours) is a constant. This means the rate of change is constant throughout the function.
• Knowing the basic rules of differentiation (power rule, constant rule) is essential.
• Don't be afraid to break down the problem step by step.

So there you have it! Finding the derivative and evaluating it at a point doesn't have to be hard. With a little practice and understanding of the basic rules, you'll be a pro in no time. Keep practicing, and you'll get the hang of it! See ya!