The Quotient Rule | Finding the Derivative of the Tangent Function (tan x)

d/dx tan x

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 if we have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) is given by:

f'(x) = (g'(x) * h(x) – g(x) * h'(x)) / (h(x))^2

In the case of tan x, we have f(x) = sin x / cos x. We can let g(x) = sin x and h(x) = cos x.

Now, let’s find the derivatives of g(x) and h(x):

g'(x) = d/dx(sin x) = cos x
h'(x) = d/dx(cos x) = -sin x

Now, substitute these derivatives into the quotient rule formula for f'(x):

f'(x) = (cos x * cos x – sin x * (-sin x)) / (cos x)^2

Simplifying the numerator:

f'(x) = (cos^2(x) + sin^2(x)) / (cos^2(x))

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify further:

f'(x) = 1 / (cos^2(x))

Since tan x = sin x / cos x, we have found that the derivative of tan x is equal to:

d/dx(tan x) = 1 / (cos^2(x))

So, the derivative of the tangent function (tan x) is equal to (1 / (cos^2(x))).

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