How To Take The Derivative Of Tan(x)

Derivative Of Tanx: In this lesson we show how to take the by-product of the tangent function incorporating the cases when the argument is a function of x. As an application, we show how this by-product is used for resembling a function.

The by-product Calculator supports gauging first, second, …, fifth derivatives as well as differentiating functions with many variables (partial derivatives), implicit differentiation and considerate roots/zeros. You can also check your answers! Interactive graphs/plots help visualize and better understand the gatherings.

For more about how to use the Derivative Calculator, go to “Help” or take a look at the examples.

To find the by-product of tangent of x, we’ll start by writing tan x as sin x/cos x and then use the quotient rule to differentiate. The quotient rule says that if two functions are differentiable, then the quotient is also differentiable. Here’s the quotient rule applied to tan x when in form of sin x/cos x:

What Is The Derivative Of Tanx

We can prove this derivative by using the by-product of sin and cos, as well as quotient rule.

Write tangent in terms of sine and cosine

Take the derivative of both sides

Use Quotient Rule

Simplify

Use the Pythagorean identity for sine and cosine

and simplify

Derivative Of Tanx^-1

Maxima takes care of actually computing the by-product of the analytical function. Like any calculator algebra system, it applies a number of rules to simplify the function and calculate the derivatives according to the commonly known discrimination rules. Maxima’s output is transformed to LaTeX again and is then presented to the user.

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 by-producthave to be guessed manually step by step. The rules of discrimination (product rule, quotient rule, chain rule, …) have been completed in JavaScript code. There is also a table of subordinate functions for the trigonometric functions and the square root, logarithm and exponential function. In each calculation step, one discrimination service is carried out or rewritten. For example, constant factors are pulled out of differentiation operations and sums are split up (sum rule). This, and general simplifications, is done by Maxima. For each calculated derivative, the LaTeX representations of the resulting mathematical expressions are identified in the HTML code so that highlighting is possible.

The “Check answer” feature has to solve the difficult task of ascertaining whether two numerical 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 aggressive 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 gatherings at randomly chosen places.

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. The gesture control is implemented using Hammer.js.

The derivative of $\tan x$ is ${\sec }^{2}x$.

To see why, you’ll need to know a few results. First, you need to know that the derivative of $\sin x$ is $\cos x$. Here’s a proof of that result from first principles:

Once you know this, it also implies that the derivative of $\cos x$ is $-\sin x$ (which you’ll also need later). You need to know one more thing, which is the Quotient Rule for differentiation:

Once all those pieces are in place, the differentiation goes as follows:

$\frac{d}{dx}\tan x$
$=\frac{d}{dx}\frac{\sin x}{\cos x}$

$=\frac{\cos x.\cos x-\sin x.(-\sin x)}{{\cos }^{2}x}$ (using Quotient Rule)

$=\frac{{\cos }^{2}x+{\sin }^{2}x}{{\cos }^{2}x}$

$=\frac{1}{{\cos }^{2}x}$ (using the Pythagorean Identity)

$={\sec }^{2}x$

Derivative Of Tanx^2

For those with a technical background, the following section explains how the Derivative Calculator works.

First, a parser analyzes the mathematical function. It transforms it into a form that is better understandable by a computer, namely a tree (see figure below). In doing this, the Derivative Calculator has to respect the order of operations. A specialty in mathematical expressions is that the multiplication sign can be left out sometimes, for example we write “5x” instead of “5*x”. The Derivative Calculator has to detect these cases and insert the multiplication sign.

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. MathJax takes care of displaying it in the browser.

When the “Go!” button is clicked, the Derivative Calculator sends the mathematical function and the settings (differentiation variable and order) to the server, where it is analyzed again. This time, the function gets transformed into a form that can be understood by the computer algebra system Maxima.

Derivative Of 1/Tanx

Proof of  sin(x) : algebraic Method

Given: lim(d->0) sin(d)/d = 1.
Solve:

 sin(x) = lim_(d->0) ( sin(x+d) – sin(x) ) / d

= lim ( sin(x)cos(d) + cos(x)sin(d) – sin(x) ) / d

= lim ( sin(x)cos(d) – sin(x) )/d + lim cos(x)sin(d)/d

= sin(x) lim ( cos(d) – 1 )/d   +   cos(x) lim sin(d)/d

= sin(x) lim ( (cos(d)-1)(cos(d)+1) ) / ( d(cos(d)+1) )   +   cos(x) lim sin(d)/d

= sin(x) lim ( cos^{^2}(d)-1 ) / ( d(cos(d)+1 )   +   cos(x) lim sin(d)/d

= sin(x) lim -sin^{^2}(d) / ( d(cos(d) + 1)   +   cos(x) lim sin(d)/d

= sin(x) lim (-sin(d)) * lim sin(d)/d * lim 1/(cos(d)+1)   +   cos(x) lim sin(d)/d

= sin(x) * 0 * 1 * 1/2 + cos(x) * 1 = cos(x)