Find the derivative of each function. $f(x)=\frac{x^{2}}{e^{x}}$

Question

Quizlet Admin · Accepted Answer

$$
\color{#c34632}	extbf{Solution:}
$$

We can solve this task using the Quotient rule since the given function is in the form of a fraction. But here we can do something else. If we observe the previous task, we can see that the function from there; denote it $g(x)=\color{#c34632}\dfrac{e^x}{x^2}$, is actually the reciprocal function according to the one given in this task.

Therefore, we have that $f(x)=\dfrac{1}{g(x)}=\dfrac{x^2}{e^x}$ and so, the derivative of our function is:

$$
\begin{align*}
f'(x)&=\Big{(}\dfrac{1}{g(x)}\Big{)}'\
&=\Big{(}(g(x))^{-1}\Big{)}'\
&=-1\cdot (g(x))^{-1-1}\cdot g'(x)	ag{	extcolor{#c34632}{Power rule and Chain rule}}\
&=-\dfrac{g'(x)}{(g(x))^2}
\end{align*}
$$

We know from the previous task that the derivative of the function $g$ is equal to $g'(x)=\dfrac{e^x\cdot (x-2)}{x^3}$, and so the required derivative of the function $f$ is:

$$
\begin{align*}
f'(x)&=-\dfrac{\frac{e^x\cdot (x-2)}{x^3}}{\frac{e^x}{x^2}}\ \
&=-\dfrac{x-2}{x}\
&=\boxed{\dfrac{2-x}{x}}
\end{align*}
$$

This derivative is defined for all real numbers for which the denominator $\color{#c34632}x$ is different from zero.
Since it is zero only for $x=0$, we have that $D_{f'}=\mathbb{R}\backslash \{0\}$.

Related questions with answers

Find the derivative of each function. $f(x)=\frac{x^{2}}{e^{x}}$

Create a free account to view solutions

Create a free account to view solutions

Recommended textbook solutions

Thomas' Calculus

Calculus: Early Transcendentals

Calculus: Early Transcendentals

Calculus for the AP Course

More related questions

Related questions with answers

Find the derivative of each function. f(x)=x2exf(x)=\frac{x^{2}}{e^{x}}f(x)=exx2​

Create a free account to view solutions

Create a free account to view solutions

Recommended textbook solutions

Thomas' Calculus

Calculus: Early Transcendentals

Calculus: Early Transcendentals

Calculus for the AP Course

More related questions

Find the derivative of each function. $f(x)=\frac{x^{2}}{e^{x}}$