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3-2 The DerivativeThurs Sept 24
Find the slope of the tangent line to y = f(x) at x = a
1) x^2 -4, a = 22) 2x^3, a = 0
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Alternative derivative notations
• There are several ways to denote the derivative. We already know f’(x)
• The following notations are all equivalent:
• These notations indicate “the derivative of y in terms of x”
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Differentiability and Continuity
• If f is differentiable at x = c (the derivative is defined at c) then f is also continuous at c
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When a function is not differentiable at a point
• When a function is not differentiable at a point x = a, the one sided limits will not be equal. There are several cases:
• A jump discontinuity (piecewise function)• Vertical asymptote• Cusp (piecewise function)• Vertical tangent line
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Derivative Info
• The derivative can tell us when a function is increasing (+), decreasing (-), or horizontal (0)
• This makes finding the vertex of a function easier
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Benefits of the Derivative
• The derivative also gives us a good view of the behavior of the original function f(x)
• Slope• Velocity• Rates of change
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Computations of Derivatives
• Thm- For any constant c,
• Note, when y = c, the slope of that line is always horizontal. Therefore, its derivative must equal 0
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• Thm- Let f(x) = x, then
• Proof:
• Note: This means that the derivative of any linear function is equal to the coefficient
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Power Rule• Let’s take a look at the different powers of x. Can you
see the pattern in the table?
F(x) F’(x)
1 0
X^1 1
X^2 2x
X^3 3x^2
X^4 4x^3
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Power Rule cont’d
Power Rule - For any real number n,
Note: The power rule works for negative exponents, as well as fraction exponents.
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Ex 3.1Find the derivatives of
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Ex 3.2Find the derivatives of
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Derivative of e^x
• The derivative of f(x) = e^x is
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General Derivative Rules
• Thm- If f(x) and g(x) are differentiable at x and c is any constant, then
• 1)
• 2)
• 3)
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General Deriv. Rules• Remember, to rewrite any expressions so
they have exponents! And split the expression into separate terms!
• You try: Find the derivative of each:• 1)
• 2)
• 3)
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Closure
• Hand in: Find the derivative of:• 1)
• 2)
• HW: p.139 #7, 13, 17, 25, 37, 43, 49