125 2.2 and 2.3
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2.2 and some of 2.3 Some Differentiation Formulas
A. Derivative of a ConstantB. Power RuleC. Evaluation of a DerivativeD. Leibniz NotationE. Derivatives in Business and Economics
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A. Derivative of a Constant
• The derivative of a constant is _______.• Examples:
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B. Power Rule
• The power comes down front with the coefficient, and you deduct one from the power. Here’s the rule:
1or
,derivative then the, If
n
n
xnxfdx
dxf
xxf
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Power Rule
• The power comes down front with the coefficient, and you deduct one from the power. Here are examples:
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What’s the derivative of x?
• What about x-5?
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What’s the derivative of x1/4?
• What’s the derivative of 8x-1/2?
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What’s the derivative of 7x?
• What’s the derivative of x3 + x5?
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What’s the derivative of x3 - x5?
• What’s the derivative of 5x-2 - 6x1/3 + 4?
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What’s the derivative of 4x-3 - 3x1/4 + 171?
• What’s the derivative of -4x1/2 - 6x-3 + x?
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C. Evaluation of a Derivative
• Evaluate the derivative of f(x)= x2 for x = 3.
• Evaluate the derivative of f(x) = x for x = 3.
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.1for 465 Evaluate 3/12 xxxdx
d
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.2 find then , If 4 fxxf
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3. at x curve theo tangent tis that line
theof slope thefind then 2x,3 If 4
xxf
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Remember when your algebra teacher taught you this:
• Rewrite this so that it has no denominators:
22 3 1x x
f xx
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The reason we need to that is …
• We don’t yet know how to differentiate
• But we know to differentiate this:
• So rewriting it will be your very first step!
22 3 1x x
f xx
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You try differentiating:
3 2
2
6 3 2 1x x xf x
x
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DON’T FORGET WHAT A DERIVATIVE IS
• It’s a function for the slope of the tangent line.• If you plug in a value for x. Let’s say, you find
the derivative of f and call it f prime. Then suppose you plug in x = 3 into f prime and get 4. So that f’(3) = 4. What does this mean?
• THE LINE THAT IS TANGENT TO f AT THE POINT x=3 HAS A SLOPE OF 4.
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D. Leibniz Notation
1
3
2
find ,4 If
wroteLeibniz ,2 writingof Instead
x
x
dx
dfxxf
dx
dff
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E. Derivatives in Business & Economics
• COST FUNCTIONS:• C(x) is a function for “total cost of producing x units.”
• MC(x) is for “MARGINAL COST” and it is the same as C’(x), the derivative of C(x).
• Marginal Cost gives you COST PER UNIT.
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REVENUE FUNCTIONS:
• R(x) is a function for the total revenue from selling x units.
• MR(x) is for “MARGINAL REVENUE,” and it is the same as R’(x), the derivative of R(x).
• Marginal Revenue gives you REVENUE PER UNIT.
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PROFIT FUNCTIONS:
• P(x) is a function for the total profit from producing and selling x units.
• Profit = Revenue minus Cost• P(x) = R(x) – C(x)
• MP(x) = P’(x) = Marginal Profit = Profit per item
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