Derivative of Tan(x): Using the Quotient Rule and Simplification

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 quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then the derivative of f(x) is given by:

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

In the case of tan(x), we can rewrite it as the ratio of sin(x) and cos(x):

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

Applying the quotient rule, we can find the derivative of tan(x) as follows:

g(x) = sin(x) and h(x) = cos(x)
g'(x) = cos(x) and h'(x) = -sin(x)

Using the quotient rule formula, we can find the derivative:

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

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

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

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

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