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2.7 Differentiation Formulas

2.7 Differentiation Formulas. We continue developing differentiation rules. Product Rule: If u ( x ) and v ( x ) are differentiable, then. Proof: Step 1. Denote u=f ( x ) , v=g ( x ) . Introduce D x , and write, as usual,

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2.7 Differentiation Formulas

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  1. 2.7 Differentiation Formulas We continue developing differentiation rules. Product Rule: If u(x)and v(x) are differentiable, then Proof: Step 1. Denote u=f(x), v=g(x). Introduce Dx, and write, as usual, u+Du=f(x+Dx), v+Dv=g(x+Dx). Now, y=uv=f(x)g(x), and y+Dy=f(x+Dx)g(x+Dx)=(u+Du)(v+Dv). Step 2. Dy=(u+Du)(v+Dv)-uv=uv+uDv+vDu+DuDv-uv= uDv+vDu+DuDv. Step 3. Step 4. 1

  2. Quotient Rule: Proof: Denote Then, u=vy, and we can differentiate it by the product rule We want to derive through u, v and their derivatives Since 2

  3. Chain Rule: If y=f(g(x)), then we introduce u=g(x), and Proof: We introduce Du=g(x+Dx)-g(x), and this is a change in function g(x), and simultaneously a chenge in the argument of the function f(u). whenever u=g(x) is continuous. Thus, Note: Generalized Power Rule is a special case of the Chain Rule. 3

  4. Example: Differentiate Chose Then Last step: substitute u as 4

  5. Exercises: Differentiate 5

  6. Homework Section 2.7: Group A: 3,7,9,11,17,25,33, Group B: 7,13,27,29. 6

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