UNIT 4.1 – Derivative of Exponential Functions
Recall:What is the definition of a derivative from first principles?Recall: Sketch the graph and the derivative of f(x)=2x24085558572500685808528000What do you notice about the shape of the graph and its derivative?Goal: We want to find an exponential function which is its own derivative!Recall that an exponential functions has the general form: fx=bx, b>0, b is a constant. Find the derivative of fx=bx using first principles. (Note: You cannot use the power rule in this case since the base is the constant!)Let’s graph the function and it’s approximate derivative on DESMOS or Geogebra. Hint: Approximate your limit for the derivative by subbing in a small value for h.For what value of b does the function and it’s derivative look the same?The exact value for b is named "e" after the mathematician who discovered it, Leonhard Euler. e=When "e" is used as the base of an exponential function, we have a function that is its own derivative!fx=ex → f'x= Example 1Find the derivative of:y=x2exb) y=ex2-x In general, if fx=egx, then f'x= Use a separate sheet of paper for the following questions:Example 2Differentiate y=exe4xExample 3Given fx=6ex2 determine f'(-1)Example 4Determine the equation of the line tangent to the graph y=xex at the point where x=2.Example 5Determine the equation of the line tangent to the graph y=exx2 , x≠0, at the point where x=2.Example 6 Find all the critical values of fx=xe-x and the associated points. Determine whether they are maximums or minimums. Example 7Differentiate fx=x5e-3x2Challenge Question Show that e2x+4e-4x>2 for all x. ................
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