# Derivatives

Unleash the power of differential calculus in the Desmos Graphing Calculator.

Plot a function and its derivative, or graph the derivative directly. Explore key concepts by building secant and tangent line sliders, or illustrate important calculus ideas like the mean value theorem.

Get started with the video on the right, then dive deeper with the resources and challenges below.

## Derivative Notation You can use $$\frac{d}{dx}$$ or $$\frac{d}{dy}$$ for derivatives. For example,$$\frac{d}{dx}$$ $$(x^{2})$$ will graph the derivative of $$x^{2}$$ with respect to $$x$$, or $$\frac{d}{dx}$$$$(sin x)$$ will graph the derivative of $$sin x$$ with respect to $$x$$. Another efficient way to implement derivative notation is by partnering it with function notation. Any defined function, like $$f(x)$$, can have its derivative graphed. You can also find $$\frac{d}{dx}$$ in the functions menu on our keyboard under the Miscellaneous tab.

## Prime Notation To use prime notation for derivatives, first try defining a function using $$f(x)$$ notation.

To enter the prime symbol, you can click on the ' button located on standard keyboards. $$f'(x)$$ can be used to graph the first order derivative of $$f(x)$$. Use $$f''(x)$$ to find the second derivative and so on. If the derivative evaluates to a constant, the value is shown in the expression list instead of on the graph.

Note that depending on the complexity of $$f(x)$$, higher order derivatives may be slow or non-existent to graph. Use prime notation to evaluate the derivative of a function at a given point.

Note that functions that are undifferentiable at given points will give a result of undefined in the expression list.

## Using Derivatives to Graph a Tangent Line You can use derivatives using either of the above notations to find the equation of a line tangent to a function in slope-intercept form. Visit this graph to see it in action.

You can replace the function used for $$f(x)$$ with any function you'd like! Notice the use of variables and sliders so you can dynamically change where the tangent line hits the function.

## Derivatives in Action

### "The best way to learn is to do." – Paul Halmos ### Interactive Tour

Master the basics of derivatives with a quick, step-by-step walkthrough. ### Graphing Challenges

Stretch your derivative skills with graphing challenges. 