Calculus 1 Lecture Notes Section 2.2 Page 5 of 6
Section 2.2: The Derivative
Big Idea: The slope of the tangent line to a function at a point is called the derivative of the function at that point.
Big Skill: You should be able to compute the derivative of a function at a point or the derivative function in general by evaluating the difference quotient limit (i.e., the limit of the secant line slopes).
Definition 2.1: The Derivative at a Point
The derivative of the function f(x) at x = a is defined as
,
provided the limit exists.
If the limit exists, we say that f is differentiable at x = a.
Alternative Definition of the Derivative at a Point
Practice:
- Find f ¢(a) for f(x) = 4x – 9 at a = 3 using both definitions of the derivative.
- Find the derivative of f(x) = 3x2 – 2x + 1 at x = -2 using both definitions of the derivative.
Definition 2.2: The Derivative Function
The derivative of the function f(x) is defined as
,
provided the limit exists.
The process of computing a derivative is called differentiation.
Notice that differentiation produces a function whose output is the derivative of the function f(x) at any value of x.
Practice:
- Find f ¢(x) for f(x) = 4x – 9.
- Find f ¢(x) for f(x) = 3x2 – 2x + 1.
- Find f ¢(x) for f(x) = 2x3 + 4x2 + x + 1.
- Find f ¢(x) for .
- Find f ¢(x) for .
- Find f ¢(x) for .
- Find f ¢(x) for .
Skill: Sketching graphs of f(x) or f ¢(x) given the other function.
Things to notice:
· Extrema of a graph have a slope / derivative of zero.
· Increasing regions of the graph have a positive derivative.
· Decreasing regions of the graph have a negative derivative.
Practice:
- Sketch the derivative of the function graphed below.
- Sketch the function whose derivative is graphed below.
Alternative derivative notations:
If y = f(x), then we also write the derivative of f(x) as:
Theorem 2.1: Relationship Between Differentiability and Continuity.
If f(x) is differentiable at x = a, then f(x) is continuous at x = a.
Proof:
Continuity at x = a requires . This is what we have to show, starting from the given that f(x) is differentiable è
QED.
This theorem captures the idea that for a derivative to exist, both one-sided limits used in the derivative must exist. Examples where the one-sided derivatives are different are functions that have sharp corners, jump discontinuities, vertical asymptotes, or cusps. The derivative also doesn’t exist where a function has a vertical tangent line, even though the function is still continuous at that point.
Practice:
Find the equation of the tangent line to the following curves at the point specified.:
- Find f ¢(0) for .
- Find f ¢(0) for .