`f(x) = tan(x)cot(x), (1,1)` Evaluate the derivative of the function at the given point. Use a graphing utility to verify your result.

Expert Answers
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You need to evaluate the equation of the tangent line to the curve f(x) = tan x*cot x, t the point (1, 1), using the following formula, such that:

`f(x) - f(1) = f'(1)(x - 1)`

Notice that f(`1` ) = 1.

You need to evaluate f'(x), using the product rule, and then f'(1):

`f'(x) = (tanx)'*cot x + (tanx)*(cot x)'`

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

Reducing like terms yields:

`f'(x) = 1/(sin x*cos x) - 1/(sin x*cos x) = 0 `

`=> f'(1) = 0`

You need to replace the values into the equation of tangent line:

`f(x) - 1 = 0*(x - 1)`

Hence, evaluating the equation of the tangent line to te given curve , at the given point, yields f(x) = 1.