Find the equation of the tangent line to the curve y=2tan(x) at the point (pi/4,2).
y'(x) = 2sec^2(x)
y'(pi/4) = 2(2) = 4
So, now you have a point and a slope, so the line is ?
To find the equation of the tangent line to the curve y = 2tan(x) at the point (π/4, 2), we need to find the slope of the tangent line and then use the point-slope form of a line to write the equation.
Step 1: Find the derivative of the function y = 2tan(x).
The derivative of tan(x) is sec^2(x), so the derivative of 2tan(x) is 2sec^2(x). Therefore, the slope of the tangent line is 2sec^2(π/4).
Step 2: Evaluate the slope of the tangent line at x = π/4.
sec(π/4) = √2^2/√2^2 = 2/√2 = √2/2.
Therefore, the slope of the tangent line at x = π/4 is 2(√2/2)^2 = 1.
Step 3: Use the point-slope form of a line to write the equation.
The equation of the tangent line can be written as (y - y1) = m(x - x1), where m is the slope and (x1, y1) is the given point.
Substituting the values, we have (y - 2) = 1(x - π/4).
Simplifying further, we get y - 2 = x - π/4.
Rearranging the equation to the standard form, we have y = x - π/4 + 2.
Therefore, the equation of the tangent line to the curve y = 2tan(x) at the point (π/4, 2) is y = x + 7π/4.