Derivative of Secx | The 3 Tools You Need

The concept of the derivative of secx, just like the derivatives of other trigonometric functions, is undoubtedly challenging to many students. Trigonometry is known for its atrocities on mathematics students. Solving for the derivative of secx will require you to be abreast of several trigonometric functions and how they relate. Tan x is an important function when solving for the derivative of secx. Our math experts have prepared this guide to help you understand the derivative of sec x and other related trigonometric problems. They also help students with their trigonometry assignments and exams.

Familiarize Yourself with the Derivative of Secx

The derivative of secx, written as d/dx (sec x) or (secx), is sec x tan x. We can apply either the quotient or the chain rule to find the derivative of sec x by referring to the fact that secx =1/cosx. The derivative of sec x can also be written in the following ways:

  • d/dn (sec n) = sec n . tan n
  • d/dp (sec p) = sec p . tan p
  • d/dk (sec k ) = sec k . tan k  

How to Approach a Problem on the Derivative of Secx

The derivative of sec x (with respect to x) is sec x tan x. Are you wondering how? Continue reading this blog and you will also find out that secx is just but the reciprocal of cos x. We will prove these results using the following methods:

  • Using Chain Rule
  • Applying Quotient Rule
  • Use of First Principle

Derivative of Secx Formula

Use the following formula to differentiate secx:

  • d/dx  (secx) = secx . tan x
  • (secx) = secx . tan x

How to Find the Derivative of Secx using the Quotient Rule

This is the ratio of the denominator quantity multiplied by the function of the numerator minus the numerator multiplied by the derivative of the denominator, all to the square root of the denominator function. You can calculate the derivative of secx using the quotient rule by following the procedure below:

dy/dx = [cos x d/dx(1) – 1 d/dx(cos x)] / (cos x)²

           = [cos x (0) – 1 (-sin x)] / (cos²x)

           = (sin x) / (cos²x)

           (1/cos x) × {(sin x)/(cos x)}

 But we know that,

   1/cos x = secx  and  sin x/cos x = tan x

Thus, after substitution we obtain,

d/dx (secx) = secx · tan x

Using the Chain Rule to Calculate the Derivative of Sec x

As for this method, you can calculate the derivative of sec x by following the procedure below:

dy/dx = (-1) (cos x)⁻² d/dx(cos x)


d/dx(cos x) = – sin x and a⁻ⁿ = 1/aⁿ


dy/dx  = -1/cos²x · (- sin x)

          = (sin x) / cos²x

         = 1/cos x · (sin x)/(cos x)

  d/dx (secx) = secx · tan x

The First Principle | An Effective Tool in Calculating the Derivative of Secx

The derivative limit of the secx by this formula can be given by:

Other Trigonometric Derivatives You Need to Know

The calculations above communicate one important message; the derivatives of trigonometric functions are interlaced. It is important that you gather general knowledge about these derivatives because it is almost impossible to solve the derivative of one function without the knowledge of the others. Notably, some of the most common derivatives of trigonometric functions that you need to know are the following:

  • Tan x – sec^(2)x
  • Cotx – (-cosec^(2)x)
  • sin^(2)x – 2sinx cosx
  • e^(cosx) – (-sinx e^(cosx))

Tips On How to Be an Ace in Trigonometric Differentiation

Trigonometric derivatives, just like other mathematical concepts, require consistency to master. This is the same case with learning how to solve the derivative of secx. What can you do to become an ace in trigonometric differentiation? The following tips are invaluable to this end:

  • Concentrate on one method and make sure you master it before moving on to another. This will help you avoid being confused about different methods.
  • If you cannot understand one method, move on to another.
  • Read extensively and do a lot of research.
  • Keep practicing the methods regularly, preferably at least twice a day.
  • Most importantly, seek help from our experts when you are stuck.

Solve for The Derivative of Sec x Like a Maestro

While finding the derivative of secx may appear arduous, you can pull off this task with the right approach. This implies that you should be comfortable with all three methods of finding this derivative: the quotient rule, the chain rule, and the first principle. Some questions require applying a specific formula, while others are open. In that case, you should pick the one you are good at. Perfection of these methods demands regular practice. Are you still facing challenges solving statistics? Contact our math experts for help. They will help you with math assignments and exams.

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