The Quotient Rule: Finding the Derivative of tan(x) with Step-by-Step Explanation

d/dx(tan(x))

To find the derivative of tan(x) with respect to x, let’s use the quotient rule

To find the derivative of tan(x) with respect to x, let’s use the quotient rule.

The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative of f(x), denoted as f'(x), can be found using the formula:

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

For the function tan(x), we have g(x) = sin(x) and h(x) = cos(x).

Now, let’s calculate the derivatives:

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

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

Applying the quotient rule:

tan'(x) = (cos(x) * cos(x) – sin(x) * (-sin(x))) / (cos(x))^2
= (cos^2(x) + sin^2(x)) / (cos^2(x))
= 1 / (cos^2(x))

Since tan(x) = sin(x)/cos(x), the derivative of tan(x), denoted as tan'(x), is equal to 1 / (cos^2(x)).

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

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