Calculating the Derivative of Tan(x) Using the Quotient Rule | Step-by-Step Guide

derivative of tanx

The derivative of the tangent function, denoted as tan(x), can be found using the quotient rule of differentiation

The derivative of the tangent function, denoted as tan(x), can be found using the quotient rule of differentiation.

Let’s start by writing down the definition of the tangent function:
tan(x) = sin(x) / cos(x)

To find the derivative of tan(x), we need to find the derivative of the numerator (sin(x)) and the derivative of the denominator (cos(x)), and then use the quotient rule.

1. Derivative of the numerator:
Using the chain rule, we know that the derivative of sin(x) is cos(x). So, d(sin(x)) / dx = cos(x).

2. Derivative of the denominator:
Similarly, the derivative of cos(x) is -sin(x). Therefore, d(cos(x)) / dx = -sin(x).

3. Apply the quotient rule:
The quotient rule states that for two functions u(x) and v(x), the derivative of their quotient, u(x) / v(x), is given by:
d(u(x) / v(x)) / dx = (v(x) * du(x) / dx – u(x) * dv(x) / dx) / (v(x))^2

Applying the quotient rule to tan(x) = sin(x) / cos(x), we have:
d(tan(x)) / dx = [(cos(x) * cos(x)) – (sin(x) * (-sin(x)))] / (cos(x))^2

Simplifying the expression further:
d(tan(x)) / dx = (cos^2(x) + sin^2(x)) / (cos^2(x))
d(tan(x)) / dx = 1 / (cos^2(x))

Therefore, the derivative of tan(x) is given by:
d(tan(x)) / dx = 1 / (cos^2(x))

Alternatively, you may also express the derivative using the identity cos^2(x) = 1 + tan^2(x), giving:
d(tan(x)) / dx = 1 / (1 + tan^2(x))

Note that this derivative is valid for all values of x such that cos(x) ≠ 0.

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