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Introduction:

The derivative is one of the most fundamental concepts in calculus. It is a measure of the rate at which a function changes, or in other words, it tells us how quickly a function is changing at any given point. This article will provide a comprehensive explanation of the derivative, including definitions, examples, and a quiz.

Definition:

The derivative of a function f(x) is defined as the rate of change of the function with respect to its independent variable, x. In other words, it is the slope of the tangent line to the curve at a given point. It can be expressed mathematically as follows:

f'(x) = lim(h->0) [(f(x+h) – f(x))/h]

This formula tells us how much the function changes as we move a small distance (h) from x. When h is very small, we get an accurate measure of the slope of the tangent line at x.

Examples:

• Consider the function f(x) = x^2. The derivative of this function can be found using the formula above:

f'(x) = lim(h->0) [(f(x+h) – f(x))/h] f'(x) = lim(h->0) [((x+h)^2 – x^2)/h] f'(x) = lim(h->0) [(x^2 + 2xh + h^2 – x^2)/h] f'(x) = lim(h->0) [2x + h] f'(x) = 2x

Therefore, the derivative of f(x) = x^2 is f'(x) = 2x.

• Consider the function g(x) = sin(x). The derivative of this function can be found using the formula above:

g'(x) = lim(h->0) [(g(x+h) – g(x))/h] g'(x) = lim(h->0) [(sin(x+h) – sin(x))/h] g'(x) = cos(x)

Therefore, the derivative of g(x) = sin(x) is g'(x) = cos(x).

• Consider the function h(x) = e^x. The derivative of this function can be found using the formula above:

h'(x) = lim(h->0) [(h(x+h) – h(x))/h] h'(x) = lim(h->0) [(e^(x+h) – e^x)/h] h'(x) = e^x

Therefore, the derivative of h(x) = e^x is h'(x) = e^x.

• Consider the function j(x) = 1/x. The derivative of this function can be found using the formula above:

j'(x) = lim(h->0) [(j(x+h) – j(x))/h] j'(x) = lim(h->0) [(1/(x+h) – 1/x)/h] j'(x) = lim(h->0) [(x – (x+h))/(x(x+h)h)] j'(x) = lim(h->0) [-1/(x(x+h))] j'(x) = -1/x^2

Therefore, the derivative of j(x) = 1/x is j'(x) = -1/x^2.

• Consider the function k(x) = x^3 + 2x^2 – 5x + 7. The derivative of this function can be found using the formula above:

k'(x) = lim(h->0) [(k(x+h) – k(x))/h] k'(x) = lim(h->0) [((x+h)^3 + 2(x+h)^2 – 5(x+h

5x + 7 – (x^3 + 2x^2 – 5x + 7))/h] k'(x) = lim(h->0) [(x^3 + 3x^2h + 3xh^2 + h^3 + 2x^2 + 4xh + 2h^2 – 5x – 5h + 7 – x^3 – 2x^2 + 5x – 7)/h] k'(x) = lim(h->0) [3x^2 + 6xh + 3h^2 + 4x – 5] k'(x) = 3x^2 + 4x – 5

Therefore, the derivative of k(x) = x^3 + 2x^2 – 5x + 7 is k'(x) = 3x^2 + 4x – 5.

Quiz:

1. What is the derivative of f(x) = x^4? A. 4x^3 B. 3x^4 C. x^3 + 4x D. x^3 + 3x^2
2. What is the derivative of g(x) = cos(x)? A. -sin(x) B. cos(x) C. tan(x) D. -cos(x)
3. What is the derivative of h(x) = ln(x)? A. 1/x B. x C. -1/x^2 D. e^x
4. What is the derivative of j(x) = 2x + 1? A. 2x B. 2 C. 1 D. 0
5. What is the derivative of k(x) = e^(2x)? A. e^2x B. 2e^x C. 2e^(2x) D. e^(2x)ln(2)
6. What is the derivative of m(x) = x/(x+1)? A. 1/(x+1) B. (x+1)^2 C. (x-1)/(x+1)^2 D. x^2/(x+1)^2
7. What is the derivative of n(x) = sqrt(x)? A. 1/2sqrt(x) B. 1/sqrt(x) C. sqrt(x) D. 2sqrt(x)
8. What is the derivative of p(x) = 1/(2x-1)? A. -1/(2x-1)^2 B. -1/(2x-1) C. 1/(2x-1)^2 D. 2/(2x-1)^2
9. What is the derivative of q(x) = sin(2x)? A. 2cos(x) B. 2cos(2x) C. cos(2x) D. 2sin(x)
10. What is the derivative of r(x) = 5x^2 – 3x + 2? A. 10x – 3 B. 10x – 2 C. 10x + 3 D. 5x^3 – 3x^2 + 2x

Answers:

1. A
2. A
3. A
4. A
5. B
6. C
7. B
8. A
9. B
10. B

Conclusion:

In conclusion, the derivative is a fundamental concept in calculus that tells us the rate at which a function is changing at any given point. It allows us to calculate the slope of a curve, find maximum and minimum points, and solve optimization problems. The process of finding the derivative is known as differentiation and it involves taking the limit of the difference quotient as the interval between two points approaches zero.

The derivative is used in a wide variety of fields including physics, economics, engineering, and finance. For example, in physics, the derivative is used to calculate velocity and acceleration of an object at any given time. In economics, it is used to calculate the marginal cost and revenue of a product. In engineering, it is used to optimize the design of structures and machines. In finance, it is used to calculate the rate of return on investments and the risk of a portfolio.

In summary, the derivative is a powerful tool in calculus that has many applications in different fields. It allows us to understand the behavior of a function at any given point and helps us solve optimization problems. By mastering the concept of the derivative, we can gain a deeper understanding of the world around us and make better decisions in our personal and professional lives.