1. Techniques of Differentiation
2.3
Techniques of Differentiation
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2. Formulas for Derivatives
f(x) & g(x) denote differentiable functions.
c,m, and b are real numbers, and n is a rational number.
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3. proof
proof
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4. Power Rule:
proof
If n is a positive integer, then we can expand by using the binomial theorem, obtaining
n-2
n-1
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5. Factor out c inside of the limit
proof
Factor out c inside of the limit
Move c out of the limit
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6. proof
proof
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7. Example 1
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8. Add and subtract the same expression
Product Rule
proof
Add and subtract the same expression
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9. Example 2
Use Product Rule
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10. Add and subtract the same expression
Quotient Rule
proof
Add and subtract the same expression
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11. Example 3

12. To prove, use the Quotient Rule with f(x)=1
Reciprocal Rule
proof
To prove, use the Quotient Rule with f(x)=1
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13. Example 4

14. Power Rule:
proof
By Reciprocal Rule:
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15. Example 2
Example 5
Solution
Simplify y
Use Power Rule
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16. Example 6 (b) Tangent is horizontal: slope = 0, or numerator = 0
Use Product Rule
(b) Tangent is horizontal: slope = 0, or numerator = 0
Tangent is vertical: slope DNE, or denominator =0
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