## d/dx(tan(x))

### The derivative of the tangent function (tan(x)) can be found using the rules of trigonometric derivatives

The derivative of the tangent function (tan(x)) can be found using the rules of trigonometric derivatives. To find the derivative of tan(x) with respect to x, 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 of f(x) with respect to x is given by:

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

In the case of tan(x), we can write it as:

f(x) = sin(x) / cos(x)

Using the quotient rule, we can find the derivative of tan(x) as follows:

f'(x) = ((sin(x))’ * cos(x) – sin(x) * (cos(x))’) / (cos(x))^2

Now, we need to differentiate sin(x) and cos(x).

The derivative of sin(x) is obtained by applying the chain rule:

(sin(x))’ = cos(x)

The derivative of cos(x) is obtained by applying the chain rule, but with a negative sign:

(cos(x))’ = -sin(x)

Substituting these results into the formula for f'(x), we get:

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

Simplifying further:

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

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the numerator:

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

Recall that 1 / cos(x) is equal to sec(x). Therefore, the derivative of tan(x) is:

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

So, the derivative of tan(x) with respect to x is sec^2(x).

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