Derivative functions

Find derivatives using their definition and check differentiability conditions.

Meaning

A derivative function assigns each input its derivative at that point. Differentiating it again gives the second derivative, which describes how the rate of change itself changes.

Example

For the derivative is At the function value is 8 and the instantaneous rate of change is 12: these are different quantities.For nonzero h, expanding ((x+h)³−x³)/h gives 3x²+3xh+h². The last two terms tend to zero with h, leaving derivative 3x².Distinguish from A derivative's units are generally the output units divided by the input units.

Practice for this concept

Scope and references

Topics

Definition of the derivative, differentiating polynomials, differentiability

Learning goal

Find derivatives using their definition and check differentiability conditions.

Helpful background

Review derivatives at a point, limits, and the function's expression.