The Quotient And Product Rules - LTCC Online

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The Product and Quotient Rules

The Product Rule

Theorem:

Let f and g be differentiable functions. Then

(f(x)g(x))' = f(x)g'(x) + f '(x)g(x)

Proof:

We have d/dx (fg) f(x+h) g(x+h) - f(x) g(x) = lim Add and subtract f(x + h)g(x) h

f(x+h) g(x+h) - f(x+h) g(x) + f(x+h) g(x) - f(x) g(x) = lim h

f(x+h) g(x+h) - f(x+h) g(x) f(x+h) g(x) - f(x) g(x) = lim + h h

g(x+h) - g(x) f(x+h) - f(x) = lim f(x+h) + g(x) h h

= [lim f(x+h)] g'(x) + g(x) f '(x)

= f(x)g'(x) + g(x)f'(x)

Example Find d (2 - x2) (x4 - 5) dx

Solution: Here f(x) = 2 - x2 and g(x) = x4 - 5 The product rule gives d/dx [f(x)g(x)] = (2 - x2)(4x3) + (-2x)(x4 - 5)

The Quotient Rule

Remember the poem

"lo d hi minus hi d lo square the bottom and away you go"

This poem is the mnemonic for the taking the derivative of a quotient.

Theorem:

d f g f ' - f g' dx g g2

Example: Find y' if 2x - 1 y = x + 1

Solution: Here f(x) = 2x - 1 and g(x) = x + 1 The quotient rule gives (x + 1) (2) - (2x - 1) (1) y' = (x + 1)2

2x + 2 - 2x + 1 = (x + 1)2

3 = (x + 1)2

Exercise

Suppose that the cost of producing x snowboards per hour is given by

50x + 1000 C = 100x + x + 2

find the marginal cost when x = 10

Answer (hold mouse over yellow rectangle for the answer)

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