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Derivative Rules

Derivative Rules. Chapter 3.3, handout. Using numeric values to calculate value of derivative at a specific x-value. “u” and “v” are functions of x that are differentiable at x = 0. u(0) = 5 v(0) = -1 u’(0) = -3 v’(0) = 2 Find d/dx(u v ) at x = 0

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Derivative Rules

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  1. Derivative Rules Chapter 3.3, handout

  2. Using numeric values to calculate value of derivative at a specific x-value • “u” and “v” are functions of x that are differentiable at x = 0. • u(0) = 5 v(0) = -1 • u’(0) = -3v’(0) = 2 • Find d/dx(uv) at x = 0 • = u(0)v’(0) + v(0)u’(0) • = 5(2) + -1(-3) = 13

  3. u” and “v” are functions of x that are differentiable at x = 0. • Find d/dx(u/v) at x = 0 if • u(0) = 5 v(0) = -1 • u’(0) = -3v’(0) = 2 • = v(0)u’(0) – u(0)v’(0) • [v(0)]2 • = -1(-3) – 5(2)= -7 • (-1)2

  4. u” and “v” are functions of x that are differentiable at x = 0. • u(0) = 5 v(0) = -1 • u’(0) = -3v’(0) = 2 Find the values of the following derivatives at x = 0: d/dx(v/u) d/dx(7v – 2u)

  5. Power Riule for Negative Integer Powers of x • If n is a negative integer and x ≠ 0, then d/dx(xn ) = nxn-1 d/dx(1/x + 2/x2) = d/dx(x-1 + 2x-2) = -x-2 – 4x-3 = -1 – 4 x2 x3 • Proof using definition of negative integer exponents and Quotient Rule for derivatives. See page 121

  6. Second and Higher Order Derivatives • A derivative, y’ = dy/dx, may itself be a differentiable function of x. Its derivative is called the second derivative of y wrt x. • y’’ = dy’/dx=d/dx(dy/dx) = d2y/dx2 If y’’ is differentiable, then its derivative is called the third derivative of y wrt x. y’’’ = dy’’/dx = d3y/dx3

  7. Higher Order Derivatives… • For fourth and higher derivatives… • y(n) = d/dx y(n-1) = dny/dxn • y(n) ≠yn “y super n” • Find the first four derivatives of y = x3 + 3x - x-1

  8. Try #35 • Find the first four derivatives of y = x2 + x-1

  9. Applications: Example 9 • Steps: • Read the problem • What rate of change (ROC) is to be determined? • Define the functions described. • Combine the functions algebraically to form a third function whose rate of change is to be determined. • Use numeric values to find the rate of change.

  10. #51 Orchard Farming • We want to determine the current rate of increase (the derivative”) of total annual production of apples • Apple production = Number of trees X Yield of apples per tree • n(t) = number of trees • y(t) = yield of apples per tree • Apple production function = a(t) = n(t)y(t)

  11. Apples!!!!!! • Apple production function = a(t) = n(t)y(t) • The derivativeof total annual production of apples is a’(t) = n(t)y’(t) + y(t)n’(t). • The current instantaneous ROC means that t = 0, so substitute numeric values: a’(0) = 156(1.5) + 12(13) = 390 bushels of apples per tree per year!

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