"What is the equation of the line tangent..." was answered by our community

Terrence Riddle

Answered question

2022-04-08

What is the equation of the line tangent to

f (x) = \sec x \tan x

x = \frac{π}{3}

posciad48o

Beginner2022-04-09Added 15 answers

f (\frac{π}{3}) = 2 \sqrt{3}

The tangent line will pass through the point

(\frac{π}{3}, 2 \sqrt{3})

.
The slope of the tangent line can be found through calculating

f^{'} (\frac{π}{3})

.
To find f'(x), use the product rule.

f^{'} (x) = \tan x \frac{d}{d x} (\sec x) + \sec x \frac{d}{d x} (\tan x)

\Rightarrow \tan x (\sec x \tan x) + \sec x (\sec^{2} x)

\Rightarrow \tan^{2} x \sec x + \sec^{3} x

f^{'} (\frac{π}{3}) = {(\sqrt{3})}^{2} (2) + {(2)}^{3} = 20

Relate the point

(\frac{π}{3}, 2 \sqrt{3})

and slope of 20 in an equation in point-slope form.

y - 2 \sqrt{3} = 20 (x - \frac{π}{3})

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