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Differentiate, Find the area of the section of the hyperbola

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Please show work step-by-step so that I can understand the process.

1. Differentiate...

4. Differentiate...

19. Find the area of the section of the hyperbola (x/a)^2 - (y/b)^2 = 1 that is bounded by the curve and the line x = 2a.

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The expert differentiates functions and sections for hyperbolas. The solution provides a complete, neat and step-by-step solution are provided in the attached file.

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1. Differentiate:

dy/dx = [1/(sec x + tan x)] * derivative of (sec x + tan x) = [1/(sec x + tan x)] * (sec x tan x + sec^2 x)
= [1/(sec x + tan x)] * sec x (sec x + tan x)
= sec x

4. Differentiate:

dy/dx = (1/2)[1/(x^2 + 2x + 2)] * derivative of (x^2 + 2x + 2) - 2 * [1/(1 + (x + 1)^2] * derivative of (x + 1)
+ [2/(x + 1)] * ...

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