# 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.

## 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 $$x$$, 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

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.

## Using Derivatives to Graph a Tangent Line

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.

## Learn More

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