Calculus I - Week 2
The Derivative: Definition and Basic Rules
This week defines the derivative rigorously using the limit definition, then builds the power rule and constant rules so students can differentiate polynomials efficiently. The derivative is introduced as both a rate of change and the slope of a tangent line.
Learning Objectives
- Write and apply the limit definition of the derivative. - Interpret the derivative as the slope of the tangent line at a point. - Interpret the derivative as an instantaneous rate of change. - Apply the constant rule and the power rule. - Differentiate any polynomial using the sum and difference rules. - Distinguish between the derivative as a function and the derivative at a point. - Use prime notation and Leibniz notation correctly.
1. The Problem That Created the Derivative
Calculus was invented to solve one fundamental problem: what is the slope of a curve at exactly one point? For a line, slope is (y2 - y1)/(x2 - x1). But a curve does not have a constant slope. The slope is different at every point. To find it, we pick two points on the curve, draw a secant line between them, and then let the second point slide toward the first. As the distance between the two points approaches zero, the secant line approaches the tangent line. That limiting slope is the derivative.
2. The Limit Definition of the Derivative
The derivative of f(x) at a point x is defined as: f'(x) = lim h->0 [f(x + h) - f(x)] / h This expression computes the slope of the secant line through the points (x, f(x)) and (x + h, f(x + h)), then takes the limit as h approaches 0. The variable h represents the horizontal distance between the two points.
3. Computing a Derivative from the Definition
Example: Find f'(x) for f(x) = x^2. Step 1. Compute f(x + h): f(x + h) = (x + h)^2 = x^2 + 2xh + h^2 Step 2. Compute the difference: f(x + h) - f(x) = x^2 + 2xh + h^2 - x^2 = 2xh + h^2 Step 3. Divide by h: (2xh + h^2) / h = 2x + h Step 4. Take the limit as h -> 0: lim h->0 (2x + h) = 2x So f'(x) = 2x.
4. Another Definition Example
Example: Find f'(x) for f(x) = x^2 + 3x. f(x + h) = (x + h)^2 + 3(x + h) = x^2 + 2xh + h^2 + 3x + 3h Difference: f(x + h) - f(x) = 2xh + h^2 + 3h Divide by h: 2x + h + 3 Limit as h -> 0: 2x + 3 So f'(x) = 2x + 3.
5. Notation: Prime and Leibniz
There are two standard notations for the derivative. Prime notation: f'(x) reads 'f prime of x.' Leibniz notation: dy/dx reads 'd-y over d-x.' It looks like a fraction but is not exactly one — it is a limit. You will see both in physics and math. Leibniz notation is especially common in physics: velocity is ds/dt, acceleration is dv/dt. At a specific point x = a, Leibniz notation writes dy/dx evaluated at a, or sometimes f'(a).
6. The Power Rule
For any real number n, if f(x) = x^n, then: f'(x) = n * x^(n-1) Examples: - f(x) = x^5 => f'(x) = 5x^4 - f(x) = x^3 => f'(x) = 3x^2 - f(x) = x => f'(x) = 1 (since x = x^1) - f(x) = 1 = x^0 => f'(x) = 0 The power rule works for negative and fractional exponents too: - f(x) = x^(-1) = 1/x => f'(x) = -x^(-2) = -1/x^2 - f(x) = x^(1/2) = sqrt(x) => f'(x) = (1/2)x^(-1/2)
7. The Constant Rule
If f(x) = c where c is any constant, then f'(x) = 0. The graph of a constant function is a horizontal line. Its slope is always 0. The constant multiple rule says: if g(x) = c * f(x), then g'(x) = c * f'(x). Example: If f(x) = 5x^3, then f'(x) = 5 * 3x^2 = 15x^2.
8. The Sum and Difference Rules
Derivatives distribute over addition and subtraction. If h(x) = f(x) + g(x), then h'(x) = f'(x) + g'(x). If h(x) = f(x) - g(x), then h'(x) = f'(x) - g'(x). This means you can differentiate a polynomial term by term. Example: f(x) = 4x^3 - 7x^2 + 2x - 9 f'(x) = 12x^2 - 14x + 2
9. Tangent Lines
The derivative at a point gives the slope of the tangent line at that point. To find the equation of the tangent line at x = a: 1. Find the slope: m = f'(a). 2. Find the point: (a, f(a)). 3. Use point-slope form: y - f(a) = m(x - a). Example: Find the tangent line to f(x) = x^2 at x = 3. f(3) = 9, so the point is (3, 9). f'(x) = 2x, so f'(3) = 6. Tangent line: y - 9 = 6(x - 3) => y = 6x - 9.
10. When a Derivative Does Not Exist
A function may not have a derivative at a point when: - The graph has a corner or sharp turn (like |x| at x = 0). - The function is discontinuous at that point. - The tangent line would be vertical (infinite slope). If the limit definition gives different values from the left and right, the derivative does not exist at that point.
11. Physics: Velocity as a Derivative
If s(t) is the position of an object at time t, then: Velocity: v(t) = s'(t) = ds/dt Acceleration: a(t) = v'(t) = dv/dt Example: If s(t) = 5t^2 - 3t + 2 (position in meters, t in seconds), then: v(t) = 10t - 3 At t = 4 seconds: v(4) = 40 - 3 = 37 meters per second.
12. Common Mistakes
- Forgetting to subtract f(x) in the numerator of the limit definition. - Applying the power rule to a constant incorrectly (the constant becomes 0, not 1). - Forgetting the coefficient when using the power rule on cx^n. - Writing dy/dx and treating it like an ordinary fraction in the limit definition. - Confusing f'(a) (a number) with f'(x) (a function).