How to Find the Derivative of the Tangent Function using the Quotient Rule

(d/dx) tan(x)

To find the derivative of the tangent function, we can use the quotient rule

To find the derivative of the tangent function, we can use the quotient rule. The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative of f(x) with respect to x is given by:

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

In this case, g(x) = sin(x) and h(x) = cos(x), so we need to find g'(x) and h'(x).

The derivative of sin(x) is cos(x), so g'(x) = cos(x).

The derivative of cos(x) is -sin(x), so h'(x) = -sin(x).

Now we can substitute g'(x), h(x), g(x), and h'(x) into the quotient rule:

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

Simplifying further:

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

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

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

Since the cosine function is equal to 1 when the tangent function is undefined (at zero or odd multiples of pi/2), the derivative of tangent is undefined at these points.

So, the derivative of tan(x) is 1/[cos(x)]^2.

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