How to Find the Derivative of the Function f(x) = tan(x) Using the Quotient Rule

Dx {tan x}=?

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

To find the derivative of the function f(x) = tan(x), we can use the quotient rule. The quotient rule states that the derivative of a quotient of two functions u(x) and v(x) is given by:

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

In this case, u(x) = sin(x) and v(x) = cos(x). The derivatives of u(x) and v(x) are u'(x) = cos(x) and v'(x) = -sin(x), respectively.

Now, we can substitute these values into the quotient rule formula:

[f(x)]’ = (cos(x) * cos(x) – sin(x) * (-sin(x))) / [cos(x)]^2

Simplifying further:

[f(x)]’ = (cos^2(x) + sin^2(x)) / cos^2(x)

Using the trigonometric identity sin^2(x) + cos^2(x) = 1:

[f(x)]’ = 1 / cos^2(x)

Recall that the reciprocal of cos(x) is sec(x):

[f(x)]’ = sec^2(x)

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

More Answers: