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# 31 - Derivatives of Polynomial& Exponential Functions...

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Unformatted text preview: Derivatives of Polynomial & Exponential Functions Given a constant function lim Given the function lim Given the function Since lim lim Given the function Since , lim lim lim lim , its derivative . lim lim , its derivative lim . lim . , its derivative lim , its derivative , MTH 173 section 3.1 notes page 1 Here's what we've seen: The Power Rule: If derivative is If is a constant and is any real number, then given the function . is a differentiable function, then , its lim If and are both differentiable, then lim lim lim lim lim Defn: is the number such that lim MTH 173 section 3.1 notes page 2 Now consider the function lim lim We have lim lim , we have by the definition The last equality occurs because, for this function, lim lim If 6. , then derivative of 10. derivative of take derivative of each term separately 16. derivative of rewrite as , then can apply the derivative rules. 24. derivative of ! rewrite as to be able to apply derivative rules MTH 173 section 3.1 notes page 3 ! ! ! ! 30 derivative of " rewite as " " ! ! ! , then ! 40 equation of tangent to rewrite function as at to apply derivative rules is slope of tangent. Tangent goes thru equation of this tangent is 44 graph in window by Sketch Calculate and graph on calculator MTH 173 section 3.1 notes page 4 derivative in blue 50 Determine equations of both lines thru Sketchpad diagram. we know the equations have the form: let the point of tangency be \$ the slope # can be calculated from defn of slope and from the derivative of the function # Solutions are The slopes are The equations are Graph to check: and . Solve to determine the values of . , using and # that are tangent to . See MTH 173 section 3.1 notes page 5 54 Determine parabola See sketchpad diagram % with tangent at of Here we know that is the point of % because tangency. and we know that the slope (read "derivative") at is because the slope of the line is . Thus we have that %, and so % % Thus we have the system of equations % % Solve to obtain Graph to check % . The parabola is MTH 173 section 3.1 notes page 6 ...
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