`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.
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.