Chapter A6, Problem 26E

To determine

To find: the derivative of y

Answer to Problem 26E

The derivative of y is $\frac{dy}{dx}=\frac{1}{\sqrt{4x^2+7x+3}}$

Explanation of Solution

Given:

  $y={\cosh }^{-1}(2\sqrt{x+1})$

Calculation:

Given $y={\cosh }^{-1}(2\sqrt{x+1})$

  $\Rightarrow y={\cosh }^{-1}\left(2{(x+1)}^{1/2}\right)$

To find $\frac{dy}{dx},$ differentiate $y$ with respect to $x$

  $\Rightarrow \frac{dy}{dx}=\frac{(2)\left(\frac{1}{2}\right){(x+1)}^{-1/2}}{\sqrt{{\left[2{(x+1)}^{1/2}\right]}^2-1}}$

  $\Rightarrow \frac{dy}{dx}=\frac{1}{\sqrt{x+1}\sqrt{4x+3}}$

  $\Rightarrow \frac{dy}{dx}=\frac{1}{\sqrt{4x^2+7x+3}}$

Conclusion:

Therefore,the derivative of y is $\Rightarrow \frac{dy}{dx}=\frac{1}{\sqrt{4x^2+7x+3}}$