Question

If anyone could help me, I'll really appreciate it.

Differentiate the following functions with respect to x.


`y = {cosh}^( - 1) (2x + 1) - {xsech}^( - 1) (x)`
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Answer 1

`(d y)/(d x) = \frac{2}{\sqrt{(2 x+1)^(2) -1} } + ((-x)/(|x|√(1-x^2)) )) + (1) Sec h^(-1) (x)`

Explanation:

Step(i):-

Given function

`y = cosh^(-1) (2 x +1) - x Sec h^(-1) (x)` ....(i)

we will use differentiation formulas

i) y = cos h⁻¹ (x)

Derivative of cos h⁻¹ (x)

`(d y)/(d x) = (1)/(√(x^2-1) )`

ii)

y = sec h⁻¹ (x)

Derivative of sec h⁻¹ (x)

`(d y)/(d x) = (-1)/(|x|√((x^2-1) )`

Apply U V formula

`(d UV)/(d x) = U V^(l) + V U^(l)`

Step(ii):-

Differentiating equation (i) with respective to 'x'

`(d y)/(d x) = \frac{1}{\sqrt{(2 x+1)^(2) -1} } X (d)/(d x) (2 x+1) + x ((-1)/(|x|√(1-x^2)) )) + (1) Sec h^(-1) (x)`

`(d y)/(d x) = \frac{1}{\sqrt{(2 x+1)^(2) -1} } X (2) + ((-x)/(|x|√(1-x^2)) )) + (1) Sec h^(-1) (x)`

Conclusion:-

`(d y)/(d x) = \frac{2}{\sqrt{(2 x+1)^(2) -1} } + ((-x)/(|x|√(1-x^2)) )) + (1) Sec h^(-1) (x)`