`f(x) = (tan(x))^2, ((pi/4),1)`Find an equation of the tangent line to the graph of f at the given point.

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Chapter 2, 2.4 - Problem 79 - Calculus of a Single Variable (10th Edition, Ron Larson).
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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to find the equation of the tangent line to the given curve, at the point `(pi/4,1)` , using the formula:

`f(x) - f(pi/4) = f'(pi/4)(x - pi/4)`

You need to notice that `f(pi/4) = 1.`

You need to evaluate the derivative of the given function, using chain rule, such that:

`f'(x) = ((tan x)^2)' => f'(x) = 2tan x*(tan x)'`

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

You need to evaluate f'(x) at `x = pi/4` , hence, you need to replace `pi/4` for x in equation of derivative:

`f'(pi/4) = 2tan (pi/4)*(1/(cos^2 (pi/4)))`

`f'(pi/4) = 2*1*1/(((sqrt2)/2)^2) => f'(pi/4) = 4`

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

`f(x) - 1 = 4(x - pi/4) => f(x) = 4x - pi + 1 `

Hence, evaluating the equation of the tangent line to the given curve, at the given point, yields `f(x) = 4x - pi + 1.`

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