An In-Depth Guide on Finding the Derivative of Tangent Function (tan x) Using Quotient Rule

derivative of tanx

To find the derivative of the tangent function (tan x), we can use the quotient rule

To find the derivative of the tangent function (tan x), we can use the quotient rule. The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient is given by:

(d/dx) [u(x) / v(x)] = [v(x) * u'(x) – u(x) * v'(x)] / [v(x)]^2

In this case, u(x) = sin x and v(x) = cos x. Taking the derivatives:

u'(x) = cos x (derivative of sin x with respect to x)
v'(x) = -sin x (derivative of cos x with respect to x)

Now, substituting these values into the quotient rule formula:

(d/dx) [tan x] = [(cos x) * sin x – (sin x) * (-sin x)] / [cos x]^2

Simplifying further:

(d/dx) [tan x] = [cos x * sin x + sin^2 x] / [cos^2 x]

Using the trigonometric identity sin^2 x + cos^2 x = 1, we can rewrite the expression as:

(d/dx) [tan x] = [cos x * sin x + (1 – cos^2 x)] / [cos^2 x]

Now, simplifying even more:

(d/dx) [tan x] = [cos x * sin x + 1 – cos^2 x] / [cos^2 x]

We can rearrange the terms and factor out cos x:

(d/dx) [tan x] = [1 – cos^2 x] / [cos^2 x] + sin x / cos x

Using the trigonometric identity 1 – cos^2 x = sin^2 x, the expression becomes:

(d/dx) [tan x] = sin^2 x / cos^2 x + sin x / cos x

Finally, using the trigonometric identity sin x / cos x = tan x, we have:

(d/dx) [tan x] = tan^2 x + tan x

Therefore, the derivative of the tangent function (tan x) is tan^2 x + tan x.

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