How to Find the Derivative of the Tan(x) Function using the Quotient Rule

Derivative of tan(x)

The derivative of tan(x) can be found using the quotient rule

The derivative of tan(x) can be found using the quotient rule. The quotient rule states that if you have a function 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 write it as:

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

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

Let g(x) = sin(x) and h(x) = cos(x).

Now, we need to find the derivatives of g(x) and h(x).

g'(x) = d/dx (sin(x)) = cos(x) (using the chain rule)

h'(x) = d/dx (cos(x)) = -sin(x) (using the chain rule)

Plugging these values into the quotient rule, 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(x)]^2

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

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

Since tan(x) is undefined when cos(x) = 0 (e.g., at x = π/2 or x = 3π/2), the derivative is also undefined at those points. Otherwise, the derivative of tan(x) is equal to 1 divided by the square of the cosine of x.

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