Chapter A6, Problem 29E

To determine

To find: the derivative of y

Answer to Problem 29E

The derivative of y is $\Rightarrow \frac{dy}{dt}=\frac{1}{2\sqrt{t}}-{\coth }^{-1}\sqrt{t}$

Explanation of Solution

Given:

  $y=(1-t){\coth }^{-1}\sqrt{t}$

Calculation:

Given $y=(1-t){\coth }^{-1}\sqrt{t}$

  $\Rightarrow y=(1-t){\coth }^{-1}\left(t^{1/2}\right)$

To find $\frac{dy}{dt},$ differentiate $y$ with respect to $t$

  $\Rightarrow \frac{dy}{dt}=(1-t)\left[\frac{(1/2)t^{-1/2}}{1-{\left(t^{1/2}\right)}^2}\right]+(-1){\coth }^{-1}\left(t^{1/2}\right)$

  $\Rightarrow \frac{dy}{dt}=\frac{1}{2\sqrt{t}}-{\coth }^{-1}\sqrt{t}$

Conclusion:

Therefore, The derivative of y is $\Rightarrow \frac{dy}{dt}=\frac{1}{2\sqrt{t}}-{\coth }^{-1}\sqrt{t}$