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Second Derivatives and Beyond

Second Derivatives and Beyond

Second Derivatives and Beyond Topics

Second Derivatives and Beyond explanations, examples, practice problems. Ready? Let’s do this.

Second Derivatives via Formulas

When we take the derivative of a differentiable function f, we end with a new function f '. If this new function f ' is differentiable, then we can take its derivative to find (f')', also...

Third Derivatives and Beyond

We're learning calculus, so, by definition, we're in the prime of our lives. Things are great. Things are awesome. We can take third derivatives, fourth derivatives, and so on. Writing f " for...


The slope or derivative of a function f describes whether f is increasing, decreasing, or constant. The concavity of a function f describes whether f is curving up, curving down, or not curving at...

Special Points

Special points are those random points awarded to the guy who invited us over to play board games. Blast you, house rules. They never seem to work in our favor. In derivative land, if we have a fun...

Extreme Points and How to Find Them

Sample ProblemThe maximum value of the function f (x) = -x2 – 1 is y = -1:Sample ProblemThe maximum value of the function f (x) = cos x is y = 1:Extreme points, also called extrema...

Local vs. Global Points

Sometimes we take vacations, sometimes we take stay-cations. Second derivatives can be used to determine if the function will be traveling somewhere extreme or if it will travel somewhere more subd...

Using Derivatives to Draw Graphs

The goal of this section is to be able to go from a formula of a function to an accurate graph of that function. We use the first and second derivatives to help find exact points on the graph and t...

In the Real World

Calculators good at graphing functions. However, calculators can only show a useful graph if a useful domain and range are specified. Sample ProblemPlay with the function f (x) = xln xIf we gr...
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