How do I find a tangent line? For example, y=2 tan (Pi x/4) at x=1

Expert Answers
Borys Shumyatskiy eNotes educator| Certified Educator


I think your function is  `y(x) = 2*tan(pi*x/4).`

First, a tangent line goes through the point at which it tangents. In our case this point is (1, y(1)), and `y(1) = 2tan(pi/4) = 2.`

Second, tangent line has slope equal to function's derivative at the point of tangency. Here

`y'(x) = 2*(1/(cos^2(pi*x/4)))*(pi/4) = (pi/2)/(cos^2(pi*x/4)).`

So   `y'(1) = (pi/2)/(cos^2(pi/4)) = (pi/2)/(1/2) = pi.`

Now we can write the equation of the tangent line.

`y-y(1) = pi*(x-1),` or

`y = pi*(x-1) + y(1) = pi*x - pi + 2.`

The answer: `y = pi*x - pi + 2.`