1.1.1 Derivatives

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Last updated 2:52 AM on 9/8/25
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4 Terms

1
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Derivative

  1. instantaneous rate of change at a specific point

  2. the slope of the tangent line to the function's graph at that point

2
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How can you define the derivative of y = f(x) at x = a using h?

  1. h represents some distance away from a

  2. meaning as h → 0, the distance is getting smaller

  3. a number divided by a smaller number makes the numerator bigger

    1. ex. 1/0.1 = 10

  4. as h gets close to zero, the numerator becomes smaller since the distance keeps decreasing

    1. ex. (7+3) - 7 = 3 → (7 + 2) - 7 = 2 → (7 + 1) - 7 = 1 → (7 + 0.0000...1) - 7 = 0

  5. Eventually, the numbers become so small that when the small denominator multiplies the small numerator, it is closer to 0 than anything else

  6. ∴ limit exists

<ol><li><p>h represents some <strong>distance</strong> away from a</p></li><li><p>meaning as h → 0, the distance is getting smaller</p></li><li><p>a number divided by a smaller number makes the numerator bigger</p><ol><li><p>ex. 1/0.1 = 10</p></li></ol></li><li><p>as h gets close to zero, the numerator becomes smaller since the <strong>distance </strong>keeps decreasing</p><ol><li><p>ex. (7+3) - 7 = 3 → (7 + 2) - 7 = 2 → (7 + 1) - 7 = 1 → (7 + 0.0000...1) - 7 = 0</p></li></ol></li><li><p>Eventually, the numbers become so small that when the small denominator multiplies the small numerator, it is closer to 0 than anything else</p></li><li><p>∴ limit exists</p></li></ol><p></p>
3
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How can you define the derivative of y = f(x) at x = a using x and a?

  1. [x, f(x)] is a point on the graph

  2. [a, f(a)] is a point on the graph

  3. as x → a, x is slowly moving closer towards a

  4. when x = a, it must have the same slope since they are the same point now

  5. the slope at x, f(x) is the slope of a, f(a)

<ol><li><p>[x, f(x)] is a point on the graph</p></li><li><p>[a, f(a)] is a point on the graph</p></li><li><p>as x → a, x is slowly moving closer towards a </p></li><li><p>when x = a, it <span style="color: red;"><strong>must</strong></span> have the same slope since they are the same point now</p></li><li><p>the slope at x, f(x) is the slope of a, f(a)</p></li></ol><p></p>
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For y = f(x) at x = a, we say that f is _____________ at a.

differentiable