Derivative of Tan(x): A Step-by-Step Guide using the Quotient Rule

d/dx(tan(x))

To find the derivative of tan(x) with respect to x, we will use the quotient rule

To find the derivative of tan(x) with respect to x, we will use the quotient rule. The quotient rule states that for a function f(x) = g(x)/h(x), the derivative f'(x) can be found as follows:

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

For the derivative of tan(x), we have:

g(x) = sin(x)
h(x) = cos(x)

Now, let’s find the derivatives of g(x) and h(x):

g'(x) = cos(x) (derivative of sin(x) is cos(x))
h'(x) = -sin(x) (derivative of cos(x) is -sin(x))

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

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

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

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

Recall from the trigonometric identity that sin^2(x) + cos^2(x) = 1. Therefore, we can simplify the above expression:

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

This is the derivative of tan(x) with respect to x.

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