Introduction to Calculus: Derivatives

Learning Objectives

Understand the concept of a limit as the foundation of the derivative.

Define the derivative as the instantaneous rate of change or the slope of the tangent line.

Calculate basic derivatives using the Power Rule.

Apply the Constant Multiple Rule and the Sum/Difference Rule for derivatives.

What is Calculus?

Calculus is the mathematical study of continuous change. It has two major branches: Differential Calculus, which is concerned with instantaneous rates of change and slopes of curves, and Integral Calculus, concerned with accumulation of quantities and areas under curves. This lesson focuses on the core concept of differential calculus: the derivative.

The Idea of a Limit

Before we can find an instantaneous rate of change, we must understand the concept of a limit. A limit describes the value that a function 'approaches' as the input approaches some value. We are interested in what happens infinitesimally close to a point, not necessarily at the point itself.

For example, the slope of a line between two points on a curve is called the secant line. To find the slope at a single point (the tangent line), we can imagine moving the two points of the secant line closer and closer together. The limit of the secant slope as the distance between the points approaches zero is the slope of the tangent line.

The Derivative

The derivative of a function f(x), denoted as f'(x) or dy/dx, gives us two important pieces of information:

1.The slope of the tangent line to the graph of f(x) at any point x.

2.The instantaneous rate of change of the function f(x) with respect to x.

For example, if a function describes an object's position over time, its derivative describes the object's instantaneous velocity.

Calculating Derivatives: The Power Rule

While the formal definition of a derivative involves limits, we can use simple rules for many common functions. The most fundamental is the Power Rule.

For any function of the form f(x) = x^n, where n is any real number, its derivative is:

f'(x) = n*x^(n-1)

In words: bring the exponent down as a multiplier, and then subtract one from the original exponent.

Examples:

If f(x) = x^3, then f'(x) = 3*x^(3-1) = 3x^2.

If f(x) = x, which is x^1, then f'(x) = 1x^(1-1) = 1x^0 = 1.

If f(x) = sqrt(x), which is x^(1/2), then f'(x) = (1/2)x^(1/2 - 1) = (1/2)x^(-1/2).

Other Basic Rules

Constant Rule: The derivative of any constant is zero. If f(x) = c, f'(x) = 0. (A horizontal line has a slope of 0).

Constant Multiple Rule: The derivative of a constant times a function is the constant times the derivative of the function. If f(x) = cg(x), then f'(x) = cg'(x).

Sum/Difference Rule: The derivative of a sum or difference of functions is the sum or difference of their derivatives. If f(x) = g(x) +/- h(x), then f'(x) = g'(x) +/- h'(x).

Putting it all together:

Find the derivative of f(x) = 5x^4 - 2x^2 + 7.

Derivative of 5x^4 is 5 * (4x^3) = 20x^3.

Derivative of -2x^2 is -2 * (2x^1) = -4x.

Derivative of 7 is 0.

So, f'(x) = 20x^3 - 4x.

Key Terms

Derivative: A measure of how a function changes as its input changes. Geometrically, it represents the slope of the line tangent to the graph of the function at a specific point.
Limit: The value that a function 'approaches' as the input 'approaches' some value. It is the foundational concept for derivatives and integrals.
Power Rule: A fundamental rule in differential calculus stating that the derivative of x^n is n*x^(n-1).
Tangent Line: A line that 'just touches' a curve at a single point, matching the curve's slope at that point. The slope of the tangent line is given by the derivative.