Exponential and Logarithmic Derivatives

Use your calculators to find the tangent line slopes to two exponential functions at x=0 y=3x y=2x mtan  0.6931 mtan  1.099 The question that arises from this is what base number would give us a slope of exactly 1 at x=0?

e  2.718 y = (2.718)x has a slope of almost 1 at x = 0 We find that the base number would be approximately…  2.718 y = (2.718)x has a slope of almost 1 at x = 0 2.718… should look familiar to you as e  2.71828 So by the derivative definition, we know that… Remember this!

Factor out an ex and we get And since we know that Which proves our easiest derivative to remember: We now have

But what about a general base derivative? First, recall these two equations from pre-calc: and Remember the chain rule and that ln a is a constant And since we can substitute ax back in for this…

Use implicit differentiation to find

And so finally we have these four derivatives to remember