Applications of Derivatives

Use derivatives for tangent lines, maxima and minima, optimization and related rates, with the setup and the first- and second-derivative tests shown. Try “maximize the area of a rectangle with perimeter 40”.

Try an example:

What you can do with the Applications of Derivatives

Tangent lines, maxima/minima, optimization and related rates.

Enter your values or describe the problem in the calculator above. Review the result and the explanation before using it in homework, study or practical calculations.

Example input

e.g. maximize the area of a rectangle with perimeter 40

How the Applications of Derivatives works

  1. Write the quantity to optimize as a function of one variable, using the constraint.
  2. Set the derivative to zero and solve for critical points.
  3. Confirm a maximum or minimum with the second derivative test and answer in context.

Frequently asked questions

How do I find a tangent line?

Find the derivative at the point for the slope, then use y − y₁ = m(x − x₁).

How do I know it is a maximum?

If f″ is negative at the critical point it is a maximum; if positive, a minimum.

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