Question

Module 3 introduces some mid-level techniques for taking derivatives of some more complicated looking functions, specifically the Product, Quotient, and Chain Rules. However, there is no "rule" that says these various rules are only used one at a time. Sometimes, we need to use more than one to arrive at a proper derivative, and that means possibly having to decide in what order to use them. Product Rule: d[f(x) · g(x)] / dx = f'(x) · g(x) + f(x) · g'(x) Quotient Rule: d[f(x) / g(x)] / dx = (f'(x) · g(x) - f(x) · g'(x)) / [g(x)]^2 , g(x) ? 0 Chain Rule: d[f(g(x))] / dx = f'(g(x)) · g'(x) or d[f(x)]^n / dx = n[f(x)]^(n-1) · f'(x) (General Power Rule) For each of the functions below, use the appropriate mixture of the above derivative rules. Break the functions into "pieces" ( f(x), f'(x), g(x), and g'(x) ) to fit into the respective formulas. Factor and/or Simplify when possible. One single term or fraction is usually possible. 1) h(x) = (2x^2 - 5)(3x + 7)^4 2) y = 8x / ?(2x + 5) 3) K(x) = (2x - 5)^3 (4x + 1)^5 4) Q(x) = (4x + 3)^5 / (6x - 1)^3 Find the derivative for this function 2 ways, utilizing Quotient Rule first, and then Product Rule the second time. Factor and simplify both to show they are indeed the same derivative.

          Module 3 introduces some mid-level techniques for taking derivatives of some more complicated looking functions, specifically the Product, Quotient, and Chain Rules. However, there is no "rule" that says these various rules are only used one at a time. Sometimes, we need to use more than one to arrive at a proper derivative, and that means possibly having to decide in what order to use them.

Product Rule: d[f(x) · g(x)] / dx = f'(x) · g(x) + f(x) · g'(x)

Quotient Rule: d[f(x) / g(x)] / dx = (f'(x) · g(x) - f(x) · g'(x)) / [g(x)]^2 , g(x) ? 0

Chain Rule: d[f(g(x))] / dx = f'(g(x)) · g'(x) or d[f(x)]^n / dx = n[f(x)]^(n-1) · f'(x) (General Power Rule)

For each of the functions below, use the appropriate mixture of the above derivative rules. Break the functions into "pieces" ( f(x), f'(x), g(x), and g'(x) ) to fit into the respective formulas.

Factor and/or Simplify when possible. One single term or fraction is usually possible.

1) h(x) = (2x^2 - 5)(3x + 7)^4

2) y = 8x / ?(2x + 5)

3) K(x) = (2x - 5)^3 (4x + 1)^5

4) Q(x) = (4x + 3)^5 / (6x - 1)^3

Find the derivative for this function 2 ways, utilizing Quotient Rule first, and then Product Rule the second time. Factor and simplify both to show they are indeed the same derivative.

$Module 3 introduces some mid-level techniques for taking derivatives of some more complicated looking functions, specifically the Product, Quotient, and Chain Rules. However, there is no "rule" that says these various rules are only used one at a time. Sometimes, we need to use more than one to arrive at a proper derivative, and that means possibly having to decide in what order to use them. Product Rule: d[f(x) · g(x)] / dx = f'(x) · g(x) + f(x) · g'(x) Quotient Rule: d[f(x) / g(x)] / dx = (f'(x) · g(x) - f(x) · g'(x)) / [g(x)]^2 , g(x) ? 0 Chain Rule: d[f(g(x))] / dx = f'(g(x)) · g'(x) or d[f(x)]^n / dx = n[f(x)]^(n-1) · f'(x) (General Power Rule) For each of the functions below, use the appropriate mixture of the above derivative rules. Break the functions into "pieces" ( f(x), f'(x), g(x), and g'(x) ) to fit into the respective formulas. Factor and/or Simplify when possible. One single term or fraction is usually possible. 1) h(x) = (2x^2 - 5)(3x + 7)^4 2) y = 8x / ?(2x + 5) 3) K(x) = (2x - 5)^3 (4x + 1)^5 4) Q(x) = (4x + 3)^5 / (6x - 1)^3 Find the derivative for this function 2 ways, utilizing Quotient Rule first, and then Product Rule the second time. Factor and simplify both to show they are indeed the same derivative.$

Added by Paula R.

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