## (tanx)’

### The derivative of the tangent function, written as (tanx)’, is determined by applying the quotient rule to the function tanx

The derivative of the tangent function, written as (tanx)’, is determined by applying the quotient rule to the function tanx.

The quotient rule states that for a function of the form f(x) = g(x)/h(x), the derivative can be found using the following formula:

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

In the case of the tangent function, we can rewrite it as tanx = sinx/cosx, where g(x) = sinx and h(x) = cosx.

Now, let’s find the derivatives of the numerator and denominator separately:

g'(x) = (sinx)’ = cosx

h'(x) = (cosx)’ = -sinx

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

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

= (cosx * cosx – sinx * (-sinx))/(cosx)^2

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

= 1/cos^2(x)

Since 1/cos^2(x) is equivalent to sec^2(x), the derivative of the tangent function is:

(tanx)’ = sec^2(x)

So, the derivative of the tangent function is simply sec^2(x). This means that for any given value of x, the rate of change of the tangent function at that point is equal to the square of the secant function evaluated at x.

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