Chapter 4.5, Problem 19E

i.

To determine

To find: To find the $dy$ for the given function $y=e^{\sin x}$ .

Answer to Problem 19E

The $dy$ for the given function $y=e^{\sin x}$ is $dy=\cos x{e}^{\sin x}dx$ .

Explanation of Solution

Given: The function $y=e^{\sin x}$

Formula used: $d(u^v)=v^{1}du$

Calculation:

The given function is $y=e^{\sin x}$

By applying the differentiation formula,

  $y=e^{\sin x}$

  $\begin{array}{l}u=e^u\\ u\text{'}=e^u\\ v=\sin x\\ v\text{'}=\cos x\end{array}$

  $dy=\cos x{e}^{\sin x}dx$

Conclusion:

The $dy$ for the given function $y=e^{\sin x}$ is $dy=\cos x{e}^{\sin x}dx$ .

ii.

To determine

To find: The value of $dy$ for the function $y=e^{\sin x}$ when $x=\pi$ and $dx=-0.1$ .

Answer to Problem 19E

The value of $dy$ for the function $y=e^{\sin x}$ when $x=\pi$ and $dx=-0.1$ is $dy=0.1$ .

Explanation of Solution

Given: The function $y=e^{\sin x}$

Formula used: $d(x^n)=n{x}^{n-1}$

Calculation:

The given function is $y=e^{\sin x}$

The differentiation of the function is,

  $dy=\cos x{e}^{\sin x}dx$

Applying $x=\pi$ and $dx=-0.1$ ,

  $\begin{array}{l}dy=\cos (\pi )e^{\sin \pi }(-0.1)\\ dy=0.1\end{array}$

Conclusion:

The value of $dy$ for the function $y=x\sqrt{1-x^2}$ when $x=\pi$ and $dx=-0.1$ is $dy=0.1$ .