Derivatives – An Introduction Concepts of primary interest: Definition: first derivative Differential of a function Left-side and right-side limits Higher derivatives Mean value and Rolle’s theorem L’Hospital Rule for indeterminate forms Implicit differentiation Inverse functions Logarithmic Derivative Method for Products Extrema and the Determinant Test for functions of two variables Lagrange Undetermined Multipliers (*** needs serious edit ****) Sample calculations: 2 Derivative of x Chain Rule Product Rule Saddle-points and local extrema Derivative of the inverse function for Squared d(sinθ) /dθ = 1 for θ = 0 Applications Closest point of approach or CPA (nautical term) Tools of the Trade Derivatives of inverse functions Contact:
[email protected] 11/1/2012 Plotting inverse functions ***ADD a table of the derivatives of common functions See Tipler; Lagrange Approximating Functions Many functions are complicated, and their evaluations require detailed, tortuous calculations. To avoid the stress, approximate representations are sometimes substituted for the actual functions of interest. For a continuous function f(x), it might be adequate to replace the function in a small interval around x0 by its value at that point f(x0). A better approximation for a function f(x) with a continuous derivative follows from the definition of that derivative. df fx()− fx ( ) = Limit 0 [DI.1] x0 xx→ dx 0 x− x0 df df Clearly, fx()≅+ fx ( ) ( x − x) where is the slope of the function at x0.. 00dx dx x0 x0 11/1/2012 Physics Handout Series.Tank: Derivatives D/DX- 2 Figure DI-1 This handout formalizes the definition of a derivative and illustrates a few basic properties and applications of the derivative.