Derivatives

Unleash the power of differential calculus in Desmos. Plot a function and its derivative, or evaluate numerical derivative values directly. Explore key concepts by building a tangent line using sliders.

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

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 \(x\), and \(x=\frac{d}{dy}\)\((\sin y)\) will graph the derivative of \(x=\sin y\) with respect to \(y\). Open this example (and any other in the article) by clicking on the image.

Another efficient way to implement derivative notation is by partnering it with function notation. With a defined function \(f(x)\) such as \(f(x)=\frac{1}{2}x^2+3x+2\), graph the derivative of \(f(x)\) with respect to \(x\) by typing \(\frac{d}{dx}(f(x))\) can have its derivative graphed.

In 3D, you can graph partial derivatives with function notation. For example, if \(f(x,y) = -x^2-y^2+2\), then \(g(x,y) = d/dx(f(x,y))\) will graph the derivative of \(f(x,y)\) with respect to \(x\) and \(h(x,y) = d/dy(f(x,y))\) will graph the derivative of \(f(x,y)\) with respect to \(y\).

Note: Depending on the complexity of your function, higher order derivatives may be slow to graph or non-existent.

Prime notation is supported for functions of a single argument. Start by defining a function in function notation such as \(f(x)=x^3+x^2+x+1\). Then, \(f’(x)\) will graph the first derivative of \(f(x)\).

Graph the second derivative with \(f’’(x)\), the third derivative with \(f’’’(x)\), and so on. If the derivative evaluates as a constant, the value is shown in the expression list instead of on the graph. Continuing with \(f(x) = x^3+x^2+x+1\), \(f’’’(x) = 6\).

Use prime notation to evaluate the derivative of a function at a given point.

If the function is undifferentiable at given points, the result will be undefined.

With functions, derivatives, and sliders, you can dynamically show that the derivative is the slope of a tangent line to the curve through any given point.

For example, if the function is \(f(x)=sin(x) + 3\) then \(f’(x)\) graphs the derivative and the equation for the tangent line through any point \((a, f(a))\) is \(y=f’(a)(x-a)+f(a)\). Click on the GIF to open the tangent line example graph and play with the slider or try out a new function.