Techniques of Differentiation

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Presentation transcript:

Techniques of Differentiation 2.3 Techniques of Differentiation Mathboat.com

Formulas for Derivatives f(x) & g(x) denote differentiable functions. c,m, and b are real numbers, and n is a rational number. Mathboat.com

proof proof Mathboat.com

Power Rule: proof If n is a positive integer, then we can expand by using the binomial theorem, obtaining n-2 n-1 Mathboat.com

Factor out c inside of the limit proof Factor out c inside of the limit Move c out of the limit Mathboat.com

proof proof Mathboat.com

Example 1 Mathboat.com

Add and subtract the same expression Product Rule proof Add and subtract the same expression Mathboat.com

Example 2 Use Product Rule Mathboat.com

Add and subtract the same expression Quotient Rule proof Add and subtract the same expression Mathboat.com

Example 3

To prove, use the Quotient Rule with f(x)=1 Reciprocal Rule proof To prove, use the Quotient Rule with f(x)=1 Mathboat.com

Example 4

Power Rule: proof By Reciprocal Rule: Mathboat.com

Example 2 Example 5 Solution Simplify y Use Power Rule Mathboat.com

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 Mathboat.com

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