Answered: "What is the second derivative of sec 2..."

parlitz3s2

Answered question

2023-01-28

What is the second derivative of

\sec^{2} x

Alissa Case

Beginner2023-01-29Added 6 answers

We must apply the Chain Rule in this situation. Let's rewrite the situation to make it simpler to understand.

\sec^{2} x

{(\sec x)}^{2}

So,

\frac{d}{d x} {(\sec x)}^{2} = 2 \sec x \cdot \frac{d}{d x} \sec x = 2 \sec x \sec x \tan x = 2 \sec^{2} x \tan x

Our first derivative is available. We now desire

\frac{d}{d x} 2 \sec^{2} x \tan x

For which we'll have to use the Product Rule and Chain Rule.

\frac{d}{d x} 2 \sec^{2} x \tan x = 2 \sec^{2} x \sec^{2} x + 4 \sec x \tan x \sec x \tan x = 2 \sec^{4} x + 4 \sec^{2} x \tan^{2} x = 2 \sec^{2} x (\sec^{2} x + 2 \tan^{2} x)

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