Exploring the Derivative of tan(x) using the Quotient Rule: Step-by-Step Guide

d/dx(tanx)

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

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

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

In the case of tan(x), we can rewrite it as the quotient of sin(x) and cos(x):

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

Therefore, the function g(x) is sin(x) and the function h(x) is cos(x).

Now, let’s find the derivative:

g'(x) = d/dx(sin(x)) = cos(x)
h'(x) = d/dx(cos(x)) = -sin(x)

By substituting these values into the quotient rule, we can find the derivative of tan(x):

f'(x) = (cos(x) * cos(x) – sin(x) * -sin(x))/(cos(x))^2
= (cos^2(x) + sin^2(x))/(cos^2(x))
= 1/(cos^2(x))

We can also rewrite this in terms of sin(x):

f'(x) = 1/(1 – sin^2(x)/cos^2(x))

Since sin^2(x)/cos^2(x) is equal to tan^2(x), we can further simplify the expression:

f'(x) = 1/(1 – tan^2(x))

This can be written as:

f'(x) = sec^2(x)

Hence, the derivative of tan(x) is sec^2(x).

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