The slope of the tangent line to the curve f(x)=4tan x at the point (π/4,4)

To find the slope of the tangent line to the curve f(x) = 4\tan x at the point \left(\frac{\pi}{4}, 4\right) , we need to find the derivative of the function and evaluate it at that point.

Step 1: Find the derivative of the function f(x) using the chain rule.
\frac{d}{dx}(4\tan x) = 4\sec^2 x

Step 2: Evaluate the derivative at the point \left(\frac{\pi}{4}, 4\right) .
\frac{d}{dx}(4\tan x) \bigg|_{x=\frac{\pi}{4}} = 4\sec^2 \left(\frac{\pi}{4}\right)

Step 3: Simplify the expression for the slope.
\sec \left(\frac{\pi}{4}\right) = \frac{1}{\cos \left(\frac{\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}

Answer: The slope of the tangent line to the curve f(x) = 4\tan x at the point \left(\frac{\pi}{4}, 4\right) is 4\left(\sqrt{2}\right)^2=8 .