BackThe Derivative as a Function 3.2
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3.2: The Derivative as a Function
Introduction to the Derivative as a Function
The derivative of a function measures the rate at which the function's value changes with respect to changes in its input. In calculus, the derivative itself is a function that assigns to each point the instantaneous rate of change at that point.
Definition: The derivative of a function f at a point a is defined as the limit:
General Derivative Function: For any x in the domain, the derivative function is:
Computing Derivatives: Examples
To compute the derivative, apply the definition to specific functions.
Example 1: For :
Example 2: For :
Application: Tangent Line to a Curve
The tangent line to the graph of a function at a point gives the best linear approximation near that point.
General Formula: The equation of the tangent line at is:
Example: For at :
Compute
Compute
Plug into the tangent line formula:
Derivative Notation
There are several common notations for derivatives:
or
or
For evaluation at :
or
or
Graphing the Derivative
The graph of the derivative function shows the slope of the original function at each point. For example, if $f(x)$ is a cubic function, $f'(x)$ will be a quadratic function.
Key Point: Where is increasing, ; where $f(x)$ is decreasing, .
Critical Points: Where has a local maximum or minimum, .
Matching Functions with Their Derivatives
Given graphs of functions and their derivatives, you can match them by analyzing the shape and slope.
Example: If a function graph is a parabola opening upwards, its derivative is a straight line with positive slope.
Matching: Use the following logic:
Function Graph | Derivative Graph |
|---|---|
(a) Increasing cubic | (D) Quadratic |
(b) S-shaped curve | (C) Double-peaked |
(c) Parabola | (B) Straight line |
(d) Decreasing line | (A) Constant |
When is a Function Not Differentiable?
A function may fail to be differentiable at certain points for several reasons:
Not Continuous: If the function has a jump or gap, it is not differentiable there.
Corner or Cusp: If the graph has a sharp point, the derivative does not exist at that point.
Vertical Tangent: If the tangent line is vertical, the derivative is undefined.
Theorem: Differentiability implies continuity. That is, if a function is differentiable at a point, it must also be continuous there.
Alternative Statement: If a function is not continuous at a point, it cannot be differentiable there.
