Introduction to Derivatives

It is not always possible to find the derivative of a function. In some cases, the derivative of a function may fail to exist at certain points on its domain, or even over its entire domain. Generally, the derivative of a function does not exist if the slope of its graph is not well-defined. In “Options” you can set the differentiation variable and the order (first, second, … derivative). You can also choose whether to show the steps and enable expression simplification.

This is because the slope to the left and right of these points are not equal. Use the limit definition of a derivative to differentiate (find the derivative of) the following functions. We can use the same method to work out derivatives of other functions (like sine, cosine, logarithms, etc). It means that, for the function x2, the slope or “rate of change” at any point is 2x.

A function that has a vertical tangent line has an infinite slope, and is therefore undefined. Suppose $$C(x)$$ is the cost (in dollars) of producing $$x$$ tons of macaroni. Functions with cusps or corners do not have defined slopes at the cusps what is an invoice and is it a legal document or corners, so they do not have derivatives at those points.

Result

Click an example to enter it into the Derivative Calculator (the current input will be deleted). Suppose $$D(t)$$ is a function that measures our distance from home (in miles) as a function of time (in hours).

In each calculation step, one differentiation operation is carried out or rewritten. For example, constant factors are pulled out of differentiation operations and sums are split up (sum rule). For each calculated derivative, the LaTeX representations of the resulting mathematical expressions are tagged in the HTML code so that highlighting is possible. Notice from the examples above that it can be fairly cumbersome to compute derivatives using the limit definition. Notice that this is beginning to look like the definition of the derivative. However, this formula gives us the slope between the two points, which is an average of the slope of the curve.

Derivatives of Other Functions

1. For example, constant factors are pulled out of differentiation operations and sums are split up (sum rule).
2. Recall that the slope of a line is the rate of change of the line, which is computed as the ratio of the change in y to the change in x.
3. Otherwise, a probabilistic algorithm is applied that evaluates and compares both functions at randomly chosen places.
4. If it can be shown that the difference simplifies to zero, the task is solved.
5. Their difference is computed and simplified as far as possible using Maxima.

This is not possible for a curve, since the slope of a curve changes from point to point. The “Check answer” feature has to solve the difficult task of determining whether two mathematical expressions are equivalent. Their difference is computed and simplified as far as possible using Maxima. For example, this involves writing trigonometric/hyperbolic functions in their exponential forms. If it can be shown that the difference simplifies to zero, the task is solved. Otherwise, a probabilistic algorithm is applied that evaluates and compares both functions at randomly chosen places.

Example: What is the derivative of cos(x)/x ?

Displaying the steps of calculation is a bit more involved, because the Derivative Calculator can’t completely depend on Maxima for this task. Instead, the derivatives have to be calculated manually step by step. The rules of differentiation (product rule, quotient rule, chain rule, …) have been implemented in JavaScript code. There is also a table of derivative functions for the trigonometric functions and the square root, logarithm and exponential function.

Maxima takes care of actually computing the derivative of the mathematical function. Like any computer algebra system, it applies a number of rules to simplify the function and calculate the derivatives according to the commonly known differentiation rules. Maxima’s output is transformed to LaTeX again and is then presented to the user. Interactive graphs/plots help visualize and better understand the functions.

Vertical tangents or infinite slope

The derivative at x is represented by the red line in the figure. To calculate the slope of this line, we need to modify the slope formula so that it can be used for a single point. We do this by computing the limit of the slope formula as the change in x (Δx), denoted h, approaches 0.

The definition of the derivative is derived from the formula for the slope of a line. Recall that the slope of a line is the rate of change of the line, which is computed as the ratio of the change in y to the change in x. Geometrically, the derivative is the slope of the line tangent to the curve at a point of interest. It is sometimes referred to as the instantaneous rate of change. Typically, we calculate the slope of a line using two points on the line.

Just like a slope tells us the direction a line is going, a derivative value tells us the direction a curve is going at a particular spot. At each point on the graph, the derivative value is the slope of the tangent line at that point. Specifically, they are slopes of lines that are tangent to the function. The Weierstrass function is continuous everywhere but differentiable nowhere! The Weierstrass function is “infinitely bumpy,” meaning that no matter how close you zoom in at any point, you will always see bumps. Therefore, you will never see a straight line with a well-defined slope no matter how much you zoom in.

The interactive function graphs are computed in the browser and displayed within a canvas element (HTML5). For each function to be graphed, the calculator creates a JavaScript function, which is then evaluated in small steps in order to draw the graph. While graphing, singularities (e.g. poles) are detected and treated specially. In the next lesson, we’ll practice differentiating functions using the definition of the derivative. The Derivative Calculator supports solving first, second…., fourth derivatives, as well as implicit differentiation and finding the zeros/roots. You can also get a better visual and understanding of the function by using our graphing tool.

When you’re done entering your function, click “Go!”, and the Derivative Calculator will show the result below. In “Examples” you will find some of the functions that are most frequently entered into the Derivative Calculator. The process of finding a derivative is called “differentiation”. The Derivative tells us the slope of a function at any point. The parser is implemented in JavaScript, based on the Shunting-yard algorithm, and can run directly in the browser. This allows for quick feedback while typing by transforming the tree into LaTeX code.