Derivatives

Updated August 20, 2021 22:20

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

Derivatives of x squared and sin of x graphed. Screenshot.

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 $x$.

$Expression line 1: f\left(x\right)=\frac{1}{2}x^{2}+3x+2. Expression line 2: \frac{d}{dx}\left(f\left(x\right)\right). The graph of f(x) and the derivative of f of x is graphed. Screenshot.$

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.

$Calculator keypad functions menu with Miscellaneous section called out. \frac{d}{dx} called out also. Screenshot.$

You can also find $\frac{d}{dx}$ in the functions menu on our keyboard under the Miscellaneous tab.

Prime Notation

$Expression line 1: f\left(x\right)=x^{3}+x^{2}+x+1. Expression line 2: f'\left(x\right). Both functions graphed. Screenshot.$

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 $f(x)$.

$Expression line 1: f\left(x\right)=x^{3}+x^{2}+x+1. Expression line 2: f'\left(x\right). Expression line 3: f''\left(x\right). Expression line 4: f'''\left(x\right) equals 6. All functions graphed. Screenshot.$

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.

$Expression line 1: f\left(x\right)=x^{3}+x^{2}+x+1. Expression line 2: f\left(0\right)= 1. Expression line 3: g\left(x\right)=x^{\frac{1}{3}}. Expression line 4: g'\left(0\right)= undefined. Screenshot.$

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.

$Expression line 1: f\left(x\right)=\frac{1}{x}. Expression line 2: g\left(x\right)=\frac{d}{dx}f\left(x\right). Expression line 3: y=g\left(a\right)\left(x-a\right)+f\left(a\right). Slider value shown at -2.2. Graph is zoomed out to include expression list and graphs. Screenshot.$

Visit this graph to see it in action.

You can replace the function used for $f(x)$ with any function you'd like!

$Expression line 1: f\left(x\right)=\frac{1}{x}. Expression line 2: g\left(x\right)=\frac{d}{dx}f\left(x\right). Expression line 3: y=g\left(a\right)\left(x-a\right)+f\left(a\right). Slider value shown at -2.2. All functions are graphed. a has a slider defined between negative ten and ten, set to a medium animation speed. Graph is zoomed out to include expression list and graphs. Animated.$