Finding the Derivative of the Tangent Function | A Step-by-Step Guide with Quotient Rule and Simplification

derivative of tangent

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

The derivative of the tangent function, denoted as tan(x), can be found using the quotient rule in calculus. The quotient rule is used to differentiate functions in the form of f(x)/g(x).

To find the derivative of tan(x), we can express it as the ratio sine(x)/cos(x). Then we can apply the quotient rule, which states that for functions u(x) = f(x)/g(x), the derivative is given by:

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

Applying this rule to the function tan(x), where f(x) = sin(x) and g(x) = cos(x), we get:

tan'(x) = [cos(x) * cos(x) – sin(x) * (-sin(x))] / [cos(x)]^2

Simplifying further:

tan'(x) = [cos^2(x) + sin^2(x)] / [cos^2(x)]

Using the identity cos^2(x) + sin^2(x) = 1, we have:

tan'(x) = 1 / [cos^2(x)]

Since sec^2(x) = 1 / cos^2(x), we can rewrite the derivative as:

tan'(x) = sec^2(x)

Therefore, the derivative of the tangent function is equal to the square of the secant function, or sec^2(x).

More Answers: