Derivative of tan x is a tool which is used to find the differentiation of tan x. Before we understand the differentiation of tan x it is important to understand about tan x. If we will describe tan x in terms of sin x and cos x, then it can be defined as the ratio of Sin x and Cos x.
Mathematically, tan x = Sin x/ Cos x
It is important to note that tan x is the reciprocal of the Cot x and vice versa. Apart from that it is also important to note that the derivative is simple denoted as d/dx. Now the question arises what is the Derivative of Tanx? So it can be simply written as d/dx (tan x).
The value of d (tan x)/dx is equal to Sec^2 x. Now in the coming context, you will learn how the derivative of tan x that is d (tan x)/dx is equal to Sec^2 x. This can be explained by the basic concepts of trigonometry and rules of derivative.
So to understand the derivative of tan x it is really important to have thorough knowledge of the trigonometry as well as the differentiation. The proof of the derivative of tan x can be done with the help of the rules of the differentiation and trigonometry which are as shown below:-
We know that tanx = Sin x/ Cosx
So the derivative of tan x, that is, d (tan x)/dx = d (Sin x/Cosx)/ dx
We know that by the divisibility rule of differentiation,
That is, d (u/v)/ d x = (v du/dx – u d v/dx)/ v^2
So d (tan x)/dx = d (Sin x/Cos x)/ dx
= (Cos x d(Sinx)/dx- Sin x d(Cosx)/dx )/ Cos ^2 x
= (Cos x. Cos x – Sin (- Sinx) ) / Cos^2 x
= ( Cos^2x + Sin^2x) / Cos ^ 2x
We know that Cos^2x + Sin^2x = 1
Therefore, d (tanx)/dx = 1/ Cos^2 x
We Know that 1/ Cos x = Sec x (This is as per the trigonometric rules)
So d (tan x)/dx = 1/ Cos^2 x = Sec^2 x (This is in line with the above equation)
Hence we have seen the proof that d/dx of tanx is equal to Sec^2 x, that is, d (tanx)/dx = 1/ Cos^2 x = Sec^2 x. But from above proof, we have learnt that without knowing the differentiation rules as well as trigonometric rules, it is impossible to prove the differentiation result of the trigonometric function.
Differentiation of one function can also help in finding the differentiation of other functions. So it is very important that we should know the differentiation of each and every function. For example sometimes the question requirement is to find the differentiation of the trigonometric as well as exponential, so in that case to find the differentiation we need to know the rules along with the individual derivative of the functions.
We can make use of Second Derivative Calculator to find out further derivatives.
Mathematically, tan x = Sin x/ Cos x
It is important to note that tan x is the reciprocal of the Cot x and vice versa. Apart from that it is also important to note that the derivative is simple denoted as d/dx. Now the question arises what is the Derivative of Tanx? So it can be simply written as d/dx (tan x).
The value of d (tan x)/dx is equal to Sec^2 x. Now in the coming context, you will learn how the derivative of tan x that is d (tan x)/dx is equal to Sec^2 x. This can be explained by the basic concepts of trigonometry and rules of derivative.
So to understand the derivative of tan x it is really important to have thorough knowledge of the trigonometry as well as the differentiation. The proof of the derivative of tan x can be done with the help of the rules of the differentiation and trigonometry which are as shown below:-
We know that tanx = Sin x/ Cosx
So the derivative of tan x, that is, d (tan x)/dx = d (Sin x/Cosx)/ dx
We know that by the divisibility rule of differentiation,
That is, d (u/v)/ d x = (v du/dx – u d v/dx)/ v^2
So d (tan x)/dx = d (Sin x/Cos x)/ dx
= (Cos x d(Sinx)/dx- Sin x d(Cosx)/dx )/ Cos ^2 x
= (Cos x. Cos x – Sin (- Sinx) ) / Cos^2 x
= ( Cos^2x + Sin^2x) / Cos ^ 2x
We know that Cos^2x + Sin^2x = 1
Therefore, d (tanx)/dx = 1/ Cos^2 x
We Know that 1/ Cos x = Sec x (This is as per the trigonometric rules)
So d (tan x)/dx = 1/ Cos^2 x = Sec^2 x (This is in line with the above equation)
Hence we have seen the proof that d/dx of tanx is equal to Sec^2 x, that is, d (tanx)/dx = 1/ Cos^2 x = Sec^2 x. But from above proof, we have learnt that without knowing the differentiation rules as well as trigonometric rules, it is impossible to prove the differentiation result of the trigonometric function.
Differentiation of one function can also help in finding the differentiation of other functions. So it is very important that we should know the differentiation of each and every function. For example sometimes the question requirement is to find the differentiation of the trigonometric as well as exponential, so in that case to find the differentiation we need to know the rules along with the individual derivative of the functions.
We can make use of Second Derivative Calculator to find out further derivatives.
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