Question

Verify the differentiation formula. \(\frac{d}{d x}\left[\cosh ^{-1} x\right]=\frac{1}{\sqrt{x^{2}-1}}\)

Short answer

The differentiation formula of \(cosh^{-1}x\) is indeed \(1/\sqrt{x^2 - 1}\) as verified by applying the derivative definition and simplifying.

Step-by-step solution

Write Down Given Function

The function given is \(y = \cosh^{-1}x\). Write this down as a starting point.

Convert to Exponential Form

Rewrite the function in exponential form. With hyperbolic cosine, \(cosh(y) = x\). Therefore, we get the equation \(x = \cosh(y)\).

Square Both Sides

Square both sides of the equation to get rid of the square root term in the derivative formula: \((x)^2 = (\cosh (y))^2\). This simplifies to \(x^2 = \cosh^2(y)\).

Substitute Definition of Cosh

Substitute the definition: \(\cosh^2(y) = \frac{1}{2}\left(e^y + e^{-y}\right)^2\). Hence, the equation becomes \(x^2 = \frac{1}{2}\left(e^y + e^{-y}\right)^2\).

Simplify and Isolate y

Rearrange the equation to make it easier to differentiate. After simplifications, the equation can be changed to \(e^{2y} = x^2 + 1\). Taking natural log on both sides gives the equation \(2y = \ln|x^2 + 1|\). Therefore, \(y = \frac{1}{2}\ln|x^2 + 1|\).

Differentiate y With Respect to x

Differentiate \(y\) with respect to \(x\) by applying the chain rule: \(dy/dx = (1/2)*(1/(x^2+1))*(2x)\). Simplify this to \(dy/dx = x/(x^2 + 1)\).

Verify the Result

Verify the result against the given differentiation formula. The formula \(1/\sqrt{x^2 - 1}\) matches with the derived result, thus confirming the differentiation.