An Introduction to Derivatives

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Derivatives.
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• The concept of a Derivative is at the core of
  Calculus and modern mathematics

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• To derive
• to take or get (something) from (something else)

For us, we are going to start with a function,
and we are going to derive another function
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• Differentiation is one of the most fundamental
  and powerful operations in all of calculus.

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It is a concept that was developed over two
hundred years ago by two people
Sir Issac Netwon (Lagrange Notation)
Gottfried Leibniz (Leibniz Notation)
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In 2 sentences.

• So far, we have been able to find the
  instantaneous rate of change (speed) at any
  point..
• The Derivative is a function that will allow us
  to calculate the instantaneous rate of change at
  every point.

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The formula for the derivative is created through
the combination of the 2 main concepts we have
studies so far1. The difference quotient2.
Limits
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• Starting with our difference quotient, we no
  longer want to construct a single secant line
  starting at a.
• We want to construct a secant line anywhere
  within the domain.
• To do this, we replace a with x

becomes
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(No Transcript)
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• Next, we no longer want this to represent a
  secant line.
• We want a tangent line.
• We want to know the exact slope at each point
  x.
• To do this, we must make h infinitely small.
• A limit will allow us to reduce h in this manner

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First Principles Definition of a Derivate The
derivative of a function f(x) is a new function
f(x) defined by
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Video of the Derivative
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Find the derivative of f(x) x2(find a function
that represents the slopes of all tangents)

• f(x)

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1

1
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Now take the limit substitute h 0 (always
try sub First)

2x
We can now sub in any value of x to determine the
slope of the tangent for every point x in the
domain..
Confirm with DESMOS!!!!!
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Other notation

• Leibniz notation
• Read as dee y by dee x
• It reminds us of the process by which the
  derivative is obtained
• D as in delta, as in the change in y with
  respect to the change in x

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Use first principles to differentiate f(x) x3

• f(x)

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The height of a javelin tossed in the air is
modelled by the function H(t) -4.9t2 10t 1,
where H is the height, and t is time, in seconds.

1. Determine the rate of change of the height of the
   javelin at time t.
2. Determine the rate of change of the javelin after
   1,2 and 3 seconds.

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• Pg 58
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