 |
VOOZH | about |
|
VOOZH | about |
Derivative of Tan x is sec2x. The derivative of Tan x refers to the process of finding the change in the tangent function with respect to the independent variable. Derivative of tan x is also known as differentiation of tan x.
In this article, we will learn about the derivative of Tan x and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule as well.
Table of Content
Among the trig derivatives, the derivative of the tan x is one of the derivatives. The derivative of the tan x is sec2x The derivative of tan x is the rate of change with respect to angle i.e. x. The resultant of the derivative of tan x is sec2x.
The formula for the derivative of tan x is given by:
(d/dx) [tan x] = sec2x
or
(tan x)’ = sec2x
Read More:
The derivative of tan x can be proved using the following ways:
To prove derivative of tan x using First Principle of Derivative, we will use basic limits and trigonometric formulas which are listed below
f'(x) = limₕ→₀ [f(x + h) - f(x)] / h ... (1)
Since f(x) = tan x, we have f(x + h) = tan (x + h).
Substituting these in (1),
f'(x) = limₕ→₀ [tan(x + h) - tan x] / h
= limₕ→₀ [ [sin (x + h) / cos (x + h)] - [sin x / cos x] ] / h
= limₕ→₀ [ [sin (x + h ) cos x - cos (x + h) sin x] / [cos x · cos(x + h)] ]/ h
We know that sin A cos B - cos A sin B = sin (A - B).
f'(x) = limₕ→₀ [ sin (x + h - x) ] / [ h cos x · cos(x + h)]
= limₕ→₀ [ sin h ] / [ h cos x · cos(x + h)]
= limₕ→₀ (sin h)/ h · limₕ→₀ 1 / [cos x · cos(x + h)]
By limit formulas, limₕ→₀ (sin h)/ h = 1.
f'(x) = 1 [ 1 / (cos x · cos(x + 0))] = 1/cos2x
since, reciprocal of cos is sec. Therefore
f'(x) = sec2x.
Hence proved.
In this we will apply quotient rule of derivative to find the formula of the derivative of tan x.
We know that
tan x = (sin x)/(cos x).
So we assume that y = (sin x)/(cos x). Then by quotient rule,
y' = [ cos x · d/dx (sin x) - sin x · d/dx (cos x)] / (cos2x)
= [cos x · cos x - sin x (-sin x)] / (cos2x)
= [cos2x + sin2x] / (cos2x)
By one of the Pythagorean identities, cos2x + sin2x = 1. So
y' = 1 / (cos2x) = sec2x
Hence proved.
In this method we will find the derivative of tan x using chain rule of derivative
For this let us assume y = tan x as y = 1 / (cot x) = (cot x)-1. Now, by using power rule and chain rule,
y' = (-1) (cot x)-2 · d/dx (cot x)
We have d/dx (cot x) = -cosec2x. Also, by a property of exponents, a-m = 1/am.
y' = -1/cot2x · (-cosec2x)
y' = tan2x · cosec2x
Now, tan x = (sin x)/(cos x) and cosec x = 1/(sin x). So
y' = (sin2x)/(cos2x) · (1/sin2x)
y' = 1/cos2x
We have 1/cos x = sec x. So
y' = sec2x
Hence proved.
Also Check:
Some examples related to Derivative of Tan x are,
Example 1: Find the derivative of tan2x
Solution:
Let f(x) = tan2x = (tan x)2
By using power rule and chain rule,
f'(x) = 2 tan x.d/dx(tan x)
We know that the derivative of tan x is sec2x
f'(x) = 2 tan x · sec2x
Hence, derivative of the given function is 2 tan x·sec2x
Example 2: Differentiate tan x with respect to sec x.
Solution:
Let us assume v = tan x and u = sec x. Then dv/dx = sec2x and du/dx = sec x · tan x.
We have to find dv/du. We can write this as
dv/du = (dv/dx) / (du/dx)
= (sec2x) / (sec x·tan x)
= (sec x) / (tan x)
= (1/cos x) / (sin x/cos x)
= 1/sin x
= cosec x
Hence, derivative of tan x with respect to sec x is cos x.
Example 3: Find the derivative of tan x·sec2x
Solution:
Let f(x) = tan x·sec2x.
By product rule,
f'(x) = tan x·d/dx (sec2x) + sec2x · d/dx(tan x)
= tan x.(2 sec x) d/dx (sec x) + sec2x (sec2x) (by chain rule)
= 2 sec x tan x (sec x tan x) + sec4x
= 2 sec2x tan2x + sec4x
Hence, derivative of the given function is 2sec2x tan2x + sec4x
Derivatives of Other Functions:
The derivative of tan(𝑥)tan(x) is sec2(𝑥)sec 2(x), and it can be derived using several methods including the limit definition, quotient rule, and chain rule. Each method provides insight into how the tangent function changes with respect to x, and sec2(𝑥) encapsulates this rate of change.
Various problems related to Derivative of Tan x are,
1. Find the derivative of tan(3x)
2. Find the derivative of tan 2x
3. Evaluate: {d}/{dx} tan(x2 + 1)
4. Evaluate the derivative of tan x.sin x
5. Find: (tan x)2.sin x
6. Evaluate the derivative of m(x)=tan(x)/x
7. Evaluate the derivative of n(x)=𝑒tan(x)
8. Evaluate the derivative of p(x)=arctan(tan(x)).
9. Evaluate the derivative of q(x)=tan(x).
10. Evaluate the derivative of r(x)=tan(x)⋅cos(x).
