## d/dx tanx

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

To find the derivative of the tangent function (tan(x)), we can use the quotient rule. Before applying the quotient rule, let’s rewrite tan(x) as sin(x)/cos(x).

The quotient rule states that if we have a function f(x) = g(x)/h(x), then the derivative f'(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 applying the quotient rule, we have:

(tan(x))’ = [(sin(x))’ * cos(x) – sin(x) * (cos(x))’] / (cos(x))^2

Now let’s find the derivatives of sin(x) and cos(x):

(sin(x))’ = cos(x)

(cos(x))’ = -sin(x)

Substituting these derivatives back into our expression:

(tan(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 simplify our answer by using the identity cos^2(x) = 1 – sin^2(x):

(tan(x))’ = 1 / (1 – sin^2(x))

Finally, we can use the identity 1 – sin^2(x) = cos^2(x) to simplify even more:

(tan(x))’ = 1 / cos^2(x) = sec^2(x)

Therefore, the derivative of tan(x) with respect to x is sec^2(x), where sec(x) represents the secant function.

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