The Derivative of the Tangent Function | Finding the Derivative using the Quotient Rule and Alternative Formulations

Derivative of Tan

The derivative of the tangent function (tanx) can be found using the quotient rule of differentiation

The derivative of the tangent function (tanx) can be found using the quotient rule of differentiation. The quotient rule states that if a function f(x) can be expressed as the ratio of two functions, u(x) and v(x), then the derivative of f(x) is given by:

f'(x) = (v(x) * u'(x) – u(x) * v'(x)) / (v(x))^2

For the tangent function, we can express it as the quotient of the sine function (sinx) and the cosine function (cosx):

tan(x) = sin(x) / cos(x)

Applying the quotient rule to find the derivative of tan(x), we have:

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)

Another way to express the derivative of the tangent function is using the identity:

tan^2(x) + 1 = sec^2(x)

By rearranging this identity, we can find an alternative form for the derivative:

tan'(x) = 1 / cos^2(x)
= sec^2(x)

Therefore, the derivative of the tangent function is equal to the secant squared function, tan'(x) = sec^2(x).

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