Derivatives

In the Desmos math tools, you can plot a function and its derivative and evaluate numerical derivative values directly.

Watch the video for an introduction to derivatives in Desmos, and read on for more information and challenges.

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 \(y\).

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

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

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 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. The equation for the tangent line through any point \((a, f(a))\) is \(y=f’(a)(x-a)+f(a)\).