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Limit Definition of Derivatives
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| f '(x1) | = |
lim Dx ® 0 |
Dy
Dx |
||
| = |
lim Dx ® 0 |
f(x1+Dx)
-f (x1)
Dx |
as the definition of the slope and derivative of f(x) at the point x = x1.
Change of Notation: In the following discussion, to make my typing easier, we will drop the subscript on x, so x1 become x, and replace Dx by h.
Definition
The slope m to a curve y = f(x) at x is defined by the limit calculation
| f¢(x) = |
lim h ® 0 |
f(x+h) -f (x) |
| Dy Dx |
= |
f(x+h) -f (x) |
For each point x = a, the value of f '(a) when defined provides the slope of the tangent line through the point [a,f(a)] or in brief at the points x = a on the curve.
Some may write
| f¢(x) = |
lim Dx ® 0 |
Dy Dx |
Reference: Limits with Parameter
In the calculation of the right hand side,
| f ¢(x) = |
lim h ® 0 |
f(x+h) -f (x) |
the limit process eliminates the dependence on h and leaves the dependence on x. The latter appears as a parameter in this calculation. Elimination of h via the limit process results in a function f '(x). We will see below how the formula for f '(x) can be obtained (derived) from the formula for f(x). For that reason, f '(x) could be called (but is not) the obtainable instead of the derivative of f(x)
Here the calculation of f '(x) give a function. If you understood, the calculations of Limits with Parameters and how the results depend on the parameter or variable x. then the following examples may be too repetitive. But the repetition is intended to suggest a pattern and to develop or confirm your algebraic reasoning skill.
Consider the function y = f(x) = x2 (again) We will compute the slope, that is the derivative of this function at x = 2, x = 3, x = 5 and x = a. Look for a pattern in the following arithmetic computations.Example 1. Let x = 2.
Then
f(2+h) - f(2) = (2+h)2-22 = 22
+ 2(2)h + h2 = 2(2)h + h2 =h(2(2) + h)
Therefore
| f '(2) = | lim h® 0 |
f(2+h)-f(2) h |
= | lim h® 0 |
h(2 + h) |
= |
lim h® 0 |
2(2) + h |
= 2(2) |
Of course 2(2) = 4, but for the sake of pattern recognition and emphasis, we keep the arithmetic expression 2(2) to the end of the calculation.
Example 2. Let x = 3. Then
f(3+h) - f(3) = (3+h)2-32 = 32
+ 2(3)h + h2 = 2(3)h + h2 =h(2(3) + h)
Therefore
| f '(3) = | lim h® 0 |
f(3+h)-f(3) h |
= | lim h® 0 |
h(3 + h) |
= |
lim h® 0 |
2(3) + h |
= 2(3) |
Of course 2(3) = 6, but for the sake of pattern recognition and emphasis, we keep the arithmetic expression 2(3) to the end of the calculation.
Example 3. Let x = 5. Then
f(5+h) - f(5) = (5+h)2-52 = 52
+ 2(5)h + h2 = 2(5)h + h2 =h(2(5) + h)
Therefore
| f '(5) = | lim h® 0 |
f(5+h)-f(5) h |
= | lim h® 0 |
h(5 + h) |
= |
lim h® 0 |
2(5) + h |
= 2(5) |
Of course 2(5) = 6, but for the sake of pattern recognition and emphasis, we keep the arithmetic expression 2(5) to the end of the calculation.
Example n. Let x = a.
Then
f(a+h) - f(a) = (a+h)2-a2 = a2
+ 2ah + h2 = 2ah + h2 =h(2a + h)
Therefore
| f '(a) = | lim h® 0 |
f(a+h)-f(a) h |
= | lim h® 0 |
h(a + h) |
= |
lim h® 0 |
2a + h |
= 2a |
Last Example. Let x = x. Then
f(x+h) - f(x) = (x+h)2-x2 = x2
+ 2xh + h2 = 2xh + h2 =h(2x + h)
Therefore
| f '(x) = | lim h® 0 |
f(x+h)-f(x) h |
= | lim h® 0 |
h(x + h) |
= |
lim h® 0 |
2x + h |
= 2x |
In the last example, we see that the derivative of f(x) = x2 is 2x. Here again the limit process eliminates the h-dependent of the slope quotient and leaves the x dependence.
[Play Video] 2¼ minutes: Derivative of x3 algebraically via Limits.
[Play Video] 2¼ minutes: Derivative as a Limit of a Quotient. First pass at finding the derivative or slope of f(x) = x2. Algebraic View.
[Play Video] 2¼ minutes: Second pass at finding the derivative or slope of f(x) = x2 at two values of x. Numerical Examples of Limit Evaluation to suggest a pattern.
The formula or definition
| f¢(x) = |
lim h ® 0 |
f(x+h) -f (x) |
provides the initial limit-based way to compute f '(x). We will use it to obtain derivatives of simple functions. But we will also introduce rules of differentiation which permit the calculation of formulas for f '(x) from formulas for f(x), calculation shortcuts for the evaluation of the limit definition of f'(x).
Here we see the plan, namely a quantity is represented by a limit. Then rules are developed to evaluate the limit directly or replace the limit evaluation by an equivalent calculation in which there is no mention of limits. But the basic properties of all these calculations come from limit considerations.
[Play Video] 2¼ minutes: Derivative of x3 algebraically via Limits.
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