What is the equation of the tangent line of f(x)=cos2xex−xtan⁡x at x=π4?

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shvatismop1rj · Accepted Answer

First, find the point the tangent line will intersect. I'll be using decimals since there is no clean way to simplify this by any means.
$f (\frac{π}{4}) = - 0.5574$
The tangent line will intersect the point $(0.7854, - 0.5574) .$
(Replacing $\frac{π}{4}$ with $0.7854 .$ )
To find the slope of the tangent line, find $f^{'} (\frac{π}{4})$ .
To find f'(x), use quotient rule for the first term and product rule for $x \tan x$ .
$f^{'} (x) = \frac{2 \cos x (- \sin x) e^{x} - e^{x} \cos^{2} x}{e^{2 x}} - \tan x - x \sec^{2} x$
$= \frac{- \cos x (2 \sin x + \cos x)}{e^{x}} - \tan x - x \sec^{2} x$
Plug in $\frac{π}{4}$ .
$f^{'} (\frac{π}{4}) = - 3.2547$
Plug the point and slope into an equation in point-slope form:
$y + 0.5574 = - 3.2547 (x - 0.7854)$

What is the equation of the tangent line

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