Derivative of cot(x)

Introduction

In this article, we will discuss a visual proof for the derivative of cot(x). cot(x) is obtained by dividing cos(x) by sin(x).

Consider a unit circle where the hypotenuse subtends an angle of x radians. The length of the opposite side AB in this scenario corresponds to the value of sin(x) and the length of the adjacent side OB has a length equal to cos(x).

Small change in angle

Now, we will increase the angle by a small value dx as shown above, this will increase the length of the opposite side, the length of the new side will be sin(x+dx) represented by CD.

Note: Before proceeding further with this article, I would highly recommend you to go through the articles explaining the derivatives of sin(x) and cos(x), if you have not already gone through them. They would give you the prerequisite understanding required to proceed further with this proof.

Changes in sin(x) and cos(x)

From the above equation we can observe that , when we increase the angle x by a small value dx the sin(x) will Increase by a length of cos(x) times dx, and cos(x) side will decrease by a length of sin(x) times dx.

Remember that dx is a very small value and cos(x) dx and sin(x) dx are also very small values. Now, using these equations we have to compare cot(x) and cot(x+dx), to get the change in the function cot(x).

Meaning of cot(x)

To visualize the meaning of cot(x) we have taken some appropriate values of sin(x) and cos(x) as lengths of lines to visualize the process of obtaining cot(x). We can observe in the above figure that cot(x) is nothing but the number of parts of sin(x) that can fit completely inside cos(x). Along with whole parts if there are some residual length, we can also use fractional parts of sin(x) to completely fill cos(x). Therefore, cot(x) is just the number of parts of sin(x) that can be placed inside cos(x).

Divide and Conquer

Visualizing cot(x+dx) is a little tricky, to make it a little easy for us, we have divided the equation into two parts by separating the numerators so that we can tackle it part by part.

First part of cot(x+dx)

In the first part of the equation, The idea is to figure out how many sin(x+dx) parts can fit into cos(x) side. In the above image you can see the length of sin(x) shown in red and the additional length shown in blue. The first thing we want to do is to arrange an equal number of sin(x+dx) parts as compared to the previous sin(x) parts in the cos(x) side. We do this to get the same cot(x) value and from there we can see how much the value increases or decreases.

Next, we can observe that since the individual sin(x+dx) parts are bigger, the combined length of the parts surpasses the length of the cos(x) side, but we don’t want this, so we need to figure out how much of these parts we need to remove. Here, we can see that the total additional length is solely due to the change in sin(x) length which is cos(x) dx times the number of parts put together.

In the above equation, the first part represents how many times the sin(x) part is inserted in the cos(x) line, the blue regions are the additional length in each part and we can observe that the number of those parts will be equal to the number of red parts, hence the second part of the equation, and each blue region has a length of cos(x) dx, therefore by multiplying the second and third parts of the equation we get the total length of the blue region which is the excess length, hence by dividing this with the denominator sin(x) we get the total value of the part that needs to be removed to make the sin(x+dx) parts to fit properly inside cos(x) side, and hence giving us the left hand side of the equation.

Second part of cot(x+dx)

The second part of the equation is much easier to simplify, The idea here is that if a small number is divided by a very large number then by making a small change to the denominator will not change the value of the division by much and the value of the division will be approximately the same.

Let us see this with an example. Take your calculator and divide 1 by 1000000, note down the result and now divide 1 by 1000001 ( which is one more than 1000000) and note down and result and compare the two. You will find the answer to be accurate to a very high accuracy. Hence, we can use the above approximated equation, since it simplifies the denominator.

Simplification

Now, bringing all the parts of the equations together and simplifying them will give us the change in the cot(x) function to be the following.

Final Equation

Finally, taking the ratio of the functional change of cot(x) and the change in the angle dx , we can arrive at the derivative of cot(x).