If you're seeing this message, it means we're having trouble loading external resources for Khan Academy.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Derivative applications
Minima, maxima, and critical points. Rates of change. Optimization. L'Hopital's rule. Mean value theorem.

Equations of normal and tangent lines

A derivative at a point in a curve can be viewed as the slope of the line tangent to that curve at that point. Given this, the natural next question is what the equation of that tangent line is. In this tutorial, we'll not only find equations of tangent lines, but normal ones as well!

Motion along a line

Derivatives can be used to calculate instantaneous rates of change. The rate of change of position with respect to time is velocity and the rate of change of velocity with respect to time is acceleration. Using these ideas, we'll be able to analyze one-dimensional particle movement given position as a function of time.

Critical points and graphing with calculus

Can calculus be used to figure out when a function takes on a local or global maximum value? Absolutely. Not only that, but derivatives and second derivatives can also help us understand the shape of the function (whether they are concave upward or downward). If you have a basic conceptual understanding of derivatives, then you can start applying that knowledge here to identify critical points, extrema, inflections points and even to graph functions.

Applying differentiation in different fields

The idea of a derivative being the instantaneous rate of change is useful when studying or thinking about phenomena in a whole range of fields. In this tutorial, we begin to just scratch the surface as we apply derivatives in fields as disperse as biology and economics.

Related rates

Have you ever wondered how fast the area of a ripple of a pond is increasing based on how fast the ripple is? What about how fast a volcano's volume is increasing? This tutorial on related rates will satiate your curiosity and then some! Solving related rates problems using calculus

Mean value theorem

If over the last hour on the highway, you averaged 60 miles per hour, then you must have been going exactly 60 miles per hour at some point. This is the gist of the mean value theorem (which generalizes the idea for any continuous, differentiable function).

L'Hôpital's rule

Limits have done their part helping to find derivatives. Now, under the guidance of l'Hôpital's rule, derivatives are looking to show their gratitude by helping to find limits. Ever try to evaluate a function at a point and get 0/0 or infinity/infinity? Well, that's a big clue that l'Hopital's rule can help you find the limit of the function at that point.