- 2 Understanding The Derivative Ap Calculus Multiple Choice
- 2 Understanding The Derivative Ap Calculus 2nd Edition
- 2 Understanding The Derivativeap Calculus 14th Edition
- 2 Understanding The Derivativeap Calculus 2nd Edition
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- 2 Understanding The Derivativeap Calculus Calculator
- Calculus derivatives If an object is dropped from a 237-foot-high building, its position (in feet above the ground) is given by d(t)= -16t^2+237, where t is the time in seconds since it was dropped.
- Derivative examples Example #1. F (x) = x 3 +5x 2 +x+8. F ' (x) = 3x 2 +2⋅5x+1+0 = 3x 2 +10x+1 Example #2. F (x) = sin(3x 2). When applying the chain rule: f ' (x.
- Calculus derivatives If an object is dropped from a 237-foot-high building, its position (in feet above the ground) is given by d(t)= -16t^2+237, where t is the time in seconds since it was dropped.
The big idea of differential calculus is the concept of the derivative, which essentially gives us the direction, or rate of change, of a function at any of its points. Learn all about derivatives and how to find them here.
It is all about slope!
Slope = Change in YChange in X |
We can find an average slope between two points. |
But how do we find the slope at a point? There is nothing to measure! |
But with derivatives we use a small difference ... ... then have it shrink towards zero. |
Let us Find a Derivative!
To find the derivative of a function y = f(x) we use the slope formula:
Slope = Change in YChange in X = ΔyΔx
And (from the diagram) we see that:
| x changes from | x | to | x+Δx |
| y changes from | f(x) | to | f(x+Δx) |
Now follow these steps:
- Fill in this slope formula: ΔyΔx = f(x+Δx) − f(x)Δx
- Simplify it as best we can
- Then make Δx shrink towards zero.
Like this:
Example: the function f(x) = x2
We know f(x) = x2, and we can calculate f(x+Δx) :
| Start with: | f(x+Δx) = (x+Δx)2 |
| Expand (x + Δx)2: | f(x+Δx) = x2 + 2x Δx + (Δx)2 |
Result: the derivative of x2 is 2x
In other words, the slope at x is 2x
We write dx instead of 'Δx heads towards 0'.
And 'the derivative of' is commonly written :
x2 = 2x
'The derivative of x2 equals 2x'
or simply 'd dx of x2 equals 2x'
What does x2 = 2x mean?
It means that, for the function x2, the slope or 'rate of change' at any point is 2x.
So when x=2 the slope is 2x = 4, as shown here:
Or when x=5 the slope is 2x = 10, and so on.
Note: sometimes f'(x) is also used for 'the derivative of':
f'(x) = 2x
'The derivative of f(x) equals 2x'
or simply 'f-dash of x equals 2x'
Let's try another example.
Example: What is x3 ?
We know f(x) = x3, and can calculate f(x+Δx) :
| Start with: | f(x+Δx) = (x+Δx)3 |
| Expand (x + Δx)3: | It actually works out to be cos2(x) − sin2(x) 2 Understanding The Derivative Ap Calculus Multiple ChoiceSo that is your next step: learn how to use the rules. 2 Understanding The Derivative Ap Calculus 2nd EditionNotation2 Understanding The Derivativeap Calculus 14th Edition'Shrink towards zero' is actually written as a limit like this:
2 Understanding The Derivativeap Calculus 2nd EditionOr sometimes the derivative is written like this (explained on Derivatives as dy/dx): 2 Understanding The Derivative Ap Calculus FrqThe process of finding a derivative is called 'differentiation'. Where to Next?2 Understanding The Derivativeap Calculus CalculatorGo and learn how to find derivatives using Derivative Rules, and get plenty of practice: |
