How To Find The Derivative Of Tan(X) Using The Quotient Rule And Secant Function

Derivative of Tan

d/dx tan(x) = sec²(x)

The derivative of tan(x) is given by:

(d/dx) tan(x) = sec^2(x)

where sec(x) is the secant function, defined as 1/cos(x).

The proof of this formula comes from the quotient rule of differentiation, which states that the derivative of the quotient of two functions is given by:

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

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

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

= (cos^2(x) * cos(x) – sin^2(x))/[cos^4(x)]

= cos(x)/cos^2(x)

= 1/cos(x)^2

= sec^2(x)

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

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