Chapter 14.4, Problem 42AYU

To determine

To find: The derivative of $f\left(x\right)=e^{-x}\sin x$ at 2.

Answer to Problem 42AYU

Solution:

$f^{\prime }\left(2\right)=-0.17928$

Explanation of Solution

Given:

$f\left(x\right)=e^{-x}\sin x$

$c=2$

Formula Used:

i. The derivative of $y=f\left(x\right)$ at $c$ is given by:

$f^{\prime }\left(c\right)=\underset{x\to c}{\lim }\frac{f\left(x\right)-f\left(c\right)}{x-c}$

ii. For any two functions $f\left(x\right)$ and $g\left(x\right)$ we have:

$\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]=f^{\prime }\left(x\right)g\left(x\right)+f\left(x\right)g^{\prime }\left(x\right)$

Calculation:

Here, $f\left(x\right)=e^{-x}\sin x$.

$f^{\prime }\left(x\right)=\frac{d}{dx}\left(e^{-x}\sin x\right)$

$f^{\prime }\left(x\right)=\frac{d}{dx}\left(e^{-x}\right)\sin x+e^{-x}\frac{d}{dx}\left(\sin x\right)$

$f^{\prime }\left(x\right)=-e^{-x}\sin x+e^{-x}\cos x$

$f^{\prime }\left(x\right)=e^{-x}\left(\cos x-\sin x\right)$

Therefore, using the graphing utility calculator we find that,

$f^{\prime }\left(2\right)=-0.17928$