Understanding the Derivative of the Tangent Function | Step-by-Step Guide to Finding the Derivative of tan(x) using the Quotient Rule and Trigonometric Identities.

tan(x) derivative

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 derive it step by step:

1. Start with the definition of the tangent function: tan(x) = sin(x) / cos(x).

2. Apply the quotient rule to find the derivative:
(d/dx) [tan(x)] = (d/dx) [sin(x) / cos(x)].

The quotient rule states: for functions f(x) = g(x) / h(x), where g(x) and h(x) are differentiable functions, the derivative is given by:
(d/dx) [f(x)] = [g'(x) * h(x) – g(x) * h'(x)] / [h(x)]^2.

3. Now, let’s find the derivatives of sin(x) and cos(x):
(d/dx) [sin(x)] = cos(x) — derivative of sin(x) is cos(x),
(d/dx) [cos(x)] = -sin(x) — derivative of cos(x) is -sin(x).

4. Apply the quotient rule by substituting into the equation from step 2:
(d/dx) [tan(x)] = [cos(x) * cos(x) – sin(x) * (-sin(x))] / [cos(x)]^2.

5. Simplify the expression:
(d/dx) [tan(x)] = [cos^2(x) + sin^2(x)] / [cos^2(x)].

6. Recall that the identity cos^2(x) + sin^2(x) is equal to 1, so we can substitute this value:
(d/dx) [tan(x)] = 1 / [cos^2(x)].

7. Finally, we can rewrite the expression using a trigonometric identity: sec^2(x) = 1 / [cos^2(x)].
Therefore, the derivative of tan(x) is:
(d/dx) [tan(x)] = sec^2(x).

To summarize, the derivative of the tangent function tan(x) is equal to the secant squared function sec^2(x).

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