Learn How to Find the Derivative of g(x) = 9 - x Using the Definition of Derivative

Calculus is a fascinating branch of mathematics that deals with the study of change and motion. One of the fundamental concepts in calculus is the derivative, which measures the rate at which a function changes. The derivative is a powerful tool that has numerous applications in science, engineering, economics, and many other fields. In this article, we will explore how to find the derivative of a function using the definition of derivative. Specifically, we will look at the function g(x) = 9 - x, which is a simple example that illustrates the basic principles of differentiation.

To begin, let us recall the definition of derivative. The derivative of a function f(x) at a point x=a is defined as the limit of the difference quotient as h approaches zero:

f'(a) = lim (h → 0) [f(a+h) - f(a)]/h

This formula may seem daunting at first, but it simply means that we are finding the slope of the tangent line to the graph of f(x) at the point (a,f(a)). To apply this definition to our function g(x) = 9 - x, we need to plug in the values of a and f(a). Let us choose a=2, so that g(2) = 9 - 2 = 7.

Using the definition of derivative, we have:

g'(2) = lim (h → 0) [g(2+h) - g(2)]/h

= lim (h → 0) [(9 - (2+h)) - 7]/h

= lim (h → 0) (7 - h)/h

At this point, we can simplify the expression by multiplying both the numerator and denominator by -1:

g'(2) = lim (h → 0) (h - 7)/(-h)

= lim (h → 0) (7 - h)/h

Now we can use algebraic manipulation to evaluate the limit. We notice that the expression (7 - h)/h is of the indeterminate form 0/0 as h approaches zero. To resolve this, we can use the technique of rationalization, which involves multiplying the numerator and denominator by the conjugate of the numerator:

g'(2) = lim (h → 0) (7 - h)/h

= lim (h → 0) [(7 - h)(7 + h)]/h(7 + h)

= lim (h → 0) (49 - h^2)/(h(7 + h))

Now we can factor out an h from the denominator:

g'(2) = lim (h → 0) (49 - h^2)/(h(7 + h))

= lim (h → 0) [(49/h) - h]/(7 + h)

Finally, we can evaluate the limit by direct substitution:

g'(2) = [(49/2) - 2]/7

= 47/14

Therefore, the derivative of g(x) = 9 - x at x=2 is g'(2) = 47/14. This means that the slope of the tangent line to the graph of g(x) at the point (2,7) is 47/14. We can use this information to find the equation of the tangent line, which is y - 7 = (47/14)(x - 2).

In conclusion, we have seen how to find the derivative of a function using the definition of derivative. Although the process may seem complex at first, it is based on simple principles of algebra and limits. By understanding the concept of derivative, we can gain insight into the behavior of functions and solve a wide range of problems in various fields.

Introduction

In calculus, the derivative of a function is a measure of how much the function changes as its input changes. The derivative is a fundamental concept in calculus and has many applications in science and engineering. In this article, we will discuss how to find the derivative of a function using the definition of derivative.

What is the Definition of Derivative?

The derivative of a function f(x) at a point x=a is defined as the limit of the difference quotient as h approaches zero:

f'(a) = lim (h → 0) [f(a + h) - f(a)] / h

This means that the derivative of a function at a point is the slope of the tangent line to the graph of the function at that point.

Example Function

Let's consider the function g(x) = 9 - x. We want to find the derivative of this function using the definition of derivative.

Step 1: Find the Difference Quotient

To find the difference quotient, we need to plug in the values of x and a into the formula:

f'(a) = lim (h → 0) [f(a + h) - f(a)] / h

For our function g(x), we have:

f'(a) = lim (h → 0) [g(a + h) - g(a)] / h

Plugging in the values of g(x) and simplifying, we get:

f'(a) = lim (h → 0) [(9 - (a + h)) - (9 - a)] / h

f'(a) = lim (h → 0) [-h] / h

f'(a) = lim (h → 0) [-1]

Step 2: Evaluate the Limit

Now we need to evaluate the limit:

f'(a) = lim (h → 0) [-1]

Since the limit is a constant, it equals the value of the function at that point:

f'(a) = -1

Result

Therefore, the derivative of the function g(x) = 9 - x is:

f'(a) = -1

This means that the slope of the tangent line to the graph of g(x) at any point is -1.

Conclusion

Finding the derivative of a function using the definition of derivative can be a straightforward process if you follow the steps correctly. It is important to understand the concept of the derivative and its applications in order to solve more complex problems in calculus and other fields. With practice, you can become proficient in finding derivatives using the definition and other methods.

Introduction to Derivatives

Derivatives are one of the most important concepts in calculus. They allow us to calculate the rate at which a function changes at any given point, and this information can be used to solve a wide variety of problems in mathematics, science, engineering, and economics. In this article, we will explore how to find the derivative of the function G(X) = 9 − X using the definition of derivative.

What is G(X) = 9 - X?

G(X) = 9 − X is a simple linear function with a slope of -1. This means that for every unit increase in X, the value of G(X) decreases by one. The graph of this function is a straight line with a y-intercept of 9 and a slope of -1, as shown below:

Definition of Derivative

The derivative of a function is a measure of how much the function changes at any given point. It is defined as the limit of the difference quotient as the change in X approaches zero:

Where f(x) is the function we want to differentiate, and h is a small increment in x, also known as the change in x.

Formula for Finding Derivative using Definition

To find the derivative of G(X) = 9 − X using the definition of derivative, we need to apply the formula above. First, we need to find the difference quotient, which is the fraction (G(X + h) - G(X))/h. Then, we need to take the limit of this fraction as h approaches zero.

Calculating the Limit

Let's begin by calculating the limit of the difference quotient using the definition of derivative. We have:

Finding the Difference Quotient

Next, we need to find the difference quotient (G(X + h) - G(X))/h. We have:

Simplifying the Difference Quotient

Now, we need to simplify the difference quotient by combining like terms and factoring out the common factor of (9 - X - h). We have:

Plugging in Values to the Formula

Next, we need to plug in the values of X and h into the formula for the derivative. We have:

Finalizing the Derivative Calculation

Finally, we need to simplify the expression above by combining like terms and taking the limit as h approaches zero. We have:

Therefore, the derivative of G(X) = 9 − X is -1.

Conclusion and Implications of the Derivative Result for G(X) = 9 - X

The derivative of G(X) = 9 − X is -1, which means that the function is decreasing at a constant rate of one unit for every unit increase in X. This information can be used to solve a wide variety of problems, such as finding the maximum or minimum value of the function, optimizing the function for a given set of constraints, or understanding the behavior of the function over time. The derivative is a powerful tool in calculus and is essential for understanding many important concepts in mathematics and science.

Finding the Derivative of G(X) = 9 − X Using the Definition of Derivative

The Story

As a math student, I always find myself struggling with calculus, especially when it comes to finding the derivative of a function. But today, I decided to confront my fear and take on the challenge of finding the derivative of G(X) = 9 − X using the definition of derivative.I started by reminding myself that the derivative of a function f(x) at a point x is defined as the limit of the difference quotient as h approaches zero. In other words, it is the slope of the tangent line to the graph of f(x) at the point x.So, I began by writing out the formula for the difference quotient:

(f(x + h) - f(x)) / h

Then, I substituted the function G(X) = 9 − X into the formula:

((9 - (x + h)) - (9 - x)) / h

The Table

To simplify things, I created a table to organize my work:| x | f(x) = 9 - x | f(x + h) = 9 - (x + h) | (f(x + h) - f(x)) / h ||------|--------------|-------------------------|-----------------------|| 1 | 8 | 7 - h | (-1 * h) / h || 0.5 | 8.5 | 8.5 - h | (-1 * h) / h || 0.1 | 8.9 | 8.9 - h | (-1 * h) / h || 0.01 | 8.99 | 8.99 - h | (-1 * h) / h |As h approaches zero, the values in the last column will approach the value of the derivative. So, I continued to simplify the expression:

((9 - (x + h)) - (9 - x)) / h = (-1 * h) / h

= -1

Therefore, the derivative of G(X) = 9 − X at any point x is equal to -1.

My Point of View

I have to admit, finding the derivative of a function using the definition of derivative can be tedious and time-consuming. However, it is an essential skill for any math student who wants to understand calculus on a deeper level. By using the formula for the difference quotient, creating a table, and simplifying the expression, I was able to find the derivative of G(X) = 9 − X with ease. I feel more confident in my ability to solve similar problems in the future.

Closing Message: Finding the Derivative Using the Definition of Derivative

In conclusion, finding the derivative using the definition of derivative may seem challenging at first, but once you understand the concept, it becomes quite simple. We started by understanding what a derivative is and why it is important in calculus. Then we moved on to the definition of derivative, which involves finding the limit as h approaches zero.

We then applied this definition to find the derivative of the function g(x) = 9 - x. We used algebraic manipulation to simplify the equation and then substituted the value of x + h into the equation. After that, we applied the limit as h approaches zero to get the final answer, which was -1.

It is important to note that this method can be applied to any function, no matter how complex it may seem. With practice, you will become more comfortable with the process and be able to find the derivative of any function using the definition of derivative.

In addition, we also discussed the importance of derivatives in real-world applications such as physics, economics, and engineering. Understanding derivatives can help us predict the behavior of various systems and make informed decisions based on that knowledge.

Lastly, I would like to encourage you to continue exploring calculus and its various concepts. There is always something new to learn, and the more you know, the more confident you will become in your abilities. Remember to practice regularly and seek help when needed.

Thank you for reading this article on finding the derivative using the definition of derivative. I hope you found it informative and helpful. Stay curious and keep learning!