## d/dx(tan x)

### To differentiate the function f(x) = tan(x) with respect to x (denoted as d/dx), we can use the quotient rule

To differentiate the function f(x) = tan(x) with respect to x (denoted as d/dx), we can use the quotient rule. The quotient rule states that if we have a function of the form f(x) = u(x)/v(x), where u(x) and v(x) are functions of x, then the derivative of f(x) is given by:

f'(x) = (u'(x)v(x) – u(x)v'(x)) / (v(x)^2)

Now, let’s differentiate f(x) = tan(x):

u(x) = sin(x)

v(x) = cos(x)

Taking the derivatives of u(x) and v(x), we get:

u'(x) = cos(x)

v'(x) = -sin(x)

Using the quotient rule, we can find the derivative of f(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)

Now, we can use the identity 1 + tan^2(x) = sec^2(x), where sec(x) represents the secant function. Rearranging this identity, we get:

tan^2(x) = sec^2(x) – 1

Dividing both sides by cos^2(x) (which is equivalent to multiplying both sides by sec^2(x)), we get:

tan^2(x) / cos^2(x) = sec^2(x) – 1 / cos^2(x)

= sec^2(x) – sec^2(x) + 1

= 1

Hence, tan^2(x) / cos^2(x) = 1. Taking the square root of both sides, we obtain:

tan(x) / cos(x) = ±1

Since cosine is never equal to zero, we can divide both sides by cos(x) and get:

tan(x) = ±1

Therefore, the derivative of tan(x) with respect to x is:

d/dx(tan(x)) = 1 / cos^2(x), or equivalently, sec^2(x)

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