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4.2:Derivatives of Products and Quotients Objectives: Students will be able to… Use and apply the product and quotient rule for differentiation
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The Product Rule The derivative of the product of 2 functions is the first function times the derivative of the second, plus the second function times the derivative of the first. Let f(x) = u(x)∙v(x)(u’(x) and v’(x) exist) f’(x) = u(x)∙v’(x) + v(x)∙u’(x) Example: Find f’(x) if f(x) = (2x+3)(x 2 -4) HOW ELSE COULD YOU HAVE DONE THIS?
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USING THE PRODUCT RULE, FIND THE DERIVATIVE OF THE FOLLOWING FUNCTIONS. 1. 2. 3. 4.
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QUOTIENT RULE The derivative of a quotient is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator. Let f(x) =, v(x) ≠ 0, v’(x) and u’(x) exist f’(x) = Be careful …Use parenthesis when subtracting function in numerator. Be aware of signs!
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Find f’(x) if f(x) =
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Use the quotient rule to find the derivatives. 1. 2.
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Find the derivative. 1. 2.
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Find the equation for the tangent line to the curve at (1,2)
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Applications of Derivative Position function: s(t) Velocity: rate of change of the position with respect to time: v(t) = s’(t) Velocity gives speed as well as direction Speed : | v(t) | Acceleration: rate of change of velocity with respect to time: a(t) = v’(t) = s’’(t)
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An object is slowing down when…. velocity and acceleration are opposite signs An object is speeding up when….. velocity and acceleration are the same sign
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