Step-by-Step Guide: Finding the Derivative of tan(x) Using the Quotient Rule

derivative of tan(x)

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

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

The tangent function, tan(x), can also be expressed as sin(x) / cos(x).

Let’s derive it step by step:

Step 1: Write the tangent function as a quotient:
tan(x) = sin(x) / cos(x)

Step 2: Apply the quotient rule:
The quotient rule states that if we have a function f(x) = g(x) / h(x), then its derivative is given by:

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

In our case, g(x) = sin(x) and h(x) = cos(x).

Step 3: Find the derivatives of g(x) and h(x):
g'(x) = cos(x) (derivative of sin(x) is cos(x))
h'(x) = -sin(x) (derivative of cos(x) is -sin(x))

Step 4: Plug these derivatives into the quotient rule formula:
tan'(x) = [cos(x) * cos(x) – sin(x) * (-sin(x))] / [cos(x)]^2
= [cos^2(x) + sin^2(x)] / cos^2(x)

Step 5: Simplify the expression:
cos^2(x) + sin^2(x) = 1 (from the Pythagorean identity)
tan'(x) = 1 / cos^2(x)

Step 6: Simplify further using the identity sec^2(x) = 1 / cos^2(x):
tan'(x) = sec^2(x)

So, the derivative of tan(x) is sec^2(x).

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