3.3 Differentiation Rules-1.pdf - 3.3 Differentiation Rules...

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1 3.3 Differentiation Rules Two Basic Derivatives Derivatives of Sums, Differences and Scalar Multiples If ? = ?(?) and ? = ?(?) are differentiable, and ? is a constant, then Examples 2: Find the derivative. 1. ?(?) = 5? 8 + 9? 3 − 7? 2 + 27 2. ? = ? 7 2 + 5 ? 3 Product and Quotient Rules: If ? and ? are differentiable functions of ? , then Example 3: Find the derivatives a) ? = (? 5 + 3)(? 2 + 8?) b) ? = 3?−4 5?+7 Examples 1 d dx (2020) = 0 d dx ( √2 π ) = 0 d dx (x 2 ) = 2x d dx (x 5 ) = 5x 4 ? ?? ( 1 ? 3 ) = ? ?? (? −3 ) = −3? −4 ? ?? (? 2 ? 2 3 ) = ? ?? (? 8/3 ) = 8 3 ? 5/3 1. Constant Functions: d dx (c) = 0 2. Power Functions: d dx (x n ) = n x n−1 1. Sum rule (? + ?) = ? + ? 2. Difference rule (? − ?) = ? − ? 3. Scalar Multiple rule (??) = ? ⋅ ? 4. Product rule (??) = ? ⋅ ? + ? ⋅ ? 5. Quotient rule ( ? ? ) = ? ⋅? − ?⋅? ? 2
2 3.3 Differentiation Rules c) ? = (2? − 7) −1 (? + 5) d) ?(𝑠) = 1−√? 1+√? The Natural Exponential Function Definition: The function ?(?) = ? ? is called the natural exponential function. Example 4: Find the derivatives of ? = ? 3 ? ? . Solution: ? = (? 3 ? ? ) = 3? 2 ? ? + ? 3 ? ? = (3? 2 + 1)?

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