Derivative of tan(x)

Introduction

In this article, we will discuss a visual proof for the derivative of tan(x). tan(x) is obtained by dividing sin(x) by cos(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 tan(x) and tan(x+dx), to get the change in the function tan(x)

Meaning of tan(x)

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

Divide and Conquer

Visualizing tan(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 tan(x+dx)

In the first part of the equation, The idea is how many parts of the cos(x+dx) can be placed inside sin(x). In the above image you can see the remaining length of cos(x) shown in red and the subtracted length shown in blue. The Red and blue parts are together arranged inside the same sin(x) line, the reason we do this is that we get the same value as tan(x) first and then use it as a reference to see how much change in the value occurs as we try to fill the remaining blue region, as you can see above and the remaining blue region when divided by cos(x) gives the partial change in the tan(x) function.

In the above equation, the first part represents the reference where we intentionally insert the same number of reduced cos(x) sides as tan(x) value, the remaining length is occupied by the blue parts which represent the uninserted length and we can observe that the number of those parts will be equal to the red parts, hence the second part of the equation, where each blue region has a length of sin(x) dx, therefore by multiplying the second and third parts of the equation we get the total length of the blue region which in not yet divided by the denominator, hence by dividing this with the denominator cos(x) we get the total value of the left hand side of the equation.

Second part of tan(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 999999 ( which is one less than 1000000) and note down and result and compare the two. You will find the answer to be accurate to a very high accuracy.

Simplification

Now, bringing all the parts of the equations together and simplifying them will give us the change in the tan(x) function to be secant squared x times dx.

Final Equation

Finally, taking the ratio of the functional change of tan(x) and the change in the angle dx , we can arrive at the derivative of tan(x), which is sec(x) squared.