Introduction to Derivatives

Derivatives are a fundamental concept in Calculus that measure the rate of change of a function. They have numerous applications in science, engineering, and economics, such as determining the slope of a tangent line, finding the velocity and acceleration of an object, and optimizing functions.

Common Terms and Definitions

Function: A rule that assigns a unique output value to each input value in its domain.

Limit: The value that a function approaches as the input approaches a specific value.

Continuous Function: A function for which small changes in the input result in small changes in the output, with no breaks or gaps in the graph.

Differentiable Function: A function that has a well-defined derivative at every point in its domain.

Tangent Line: A straight line that touches a curve at a single point without crossing it.

Secant Line: A straight line that intersects a curve at two points.

Definition of the Derivative

The derivative of a function f(x) at a point x, denoted as f'(x), is defined as the limit of the difference quotient as h approaches 0:

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

This definition captures the idea of the instantaneous rate of change of the function at the point x.

Rules for Finding Derivatives

There are several rules and techniques for finding derivatives:

1. Constant Rule: The derivative of a constant is 0.
2. Power Rule: For a function f(x) = x^n, the derivative is f'(x) = n * x^(n-1).
3. Sum Rule: The derivative of a sum of functions is the sum of their derivatives.
4. Product Rule: For functions f(x) and g(x), the derivative of their product is f'(x)g(x) + f(x)g'(x).
5. Quotient Rule: For functions f(x) and g(x), the derivative of their quotient is (g(x)f'(x) - f(x)g'(x)) / (g(x))^2.
6. Chain Rule: For a composite function f(g(x)), the derivative is f'(g(x)) * g'(x).

Applications of Derivatives

Derivatives have numerous applications, including:

• Finding the slope of a tangent line to a curve at a given point.
• Determining the rate of change of a function, such as velocity and acceleration in physics.
• Optimizing functions to find maximum or minimum values, which is useful in economics and engineering.
• Approximating functions using linear approximations and Taylor series.

Common Questions and Answers

What is the difference between a secant line and a tangent line?

A secant line intersects a curve at two points, while a tangent line touches a curve at a single point without crossing it. As the two points of intersection of a secant line approach each other, the secant line approaches the tangent line.

How do you find the equation of a tangent line to a curve at a given point?

To find the equation of a tangent line, first find the derivative of the function at the given point. The slope of the tangent line is equal to the value of the derivative at that point. Then, use the point-slope form of a line equation to write the equation of the tangent line.

What is the relationship between differentiability and continuity?

If a function is differentiable at a point, it must be continuous at that point. However, a function can be continuous at a point without being differentiable there. For example, the absolute value function f(x) = |x| is continuous at x = 0 but not differentiable at that point.

Conclusion

Derivatives are a crucial concept in Calculus, with a wide range of applications in various fields. By understanding the definition, rules, and techniques for finding derivatives, as well as their applications, you will be well-equipped to solve problems involving rates of change and optimization. Practice is key to mastering derivatives, so be sure to work through plenty of examples and problems to reinforce your understanding.