Derivative of sec(x)

Introduction

In this article, we are going to discuss the visual derivation of the Derivative of sec(x). sec(x) is defined as one over cos(x) or in general it is the ratio of the hypotenuse and the adjacent side to the angle, in a right angled triangle.

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

Small change in angle

Now, we will increase the angle by a small value dx as shown above, this will decrease the length of the adjacent side, the length of the new side will be cos(x+dx) represented by OD.

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.

Change in cos(x)

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

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

Meaning of sec(x)

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

Change in sec(x)

The idea is how many parts of the cos(x+dx) can be placed inside Hypotenuse. 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 hypontenuse line, the reason we do this is that we get the same value as sec(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 sec(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 sec(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 change in the value of the sec(x).

Simplification

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

Derivative of sec(x)

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