Question

chain rule
question
part 1 of 2
my score: 8.67/20 pts
write the function in the form y = f(u) and u = g(x). then find \\(\frac{dy}{dx}\\) as a function of x.
y = e^{-21x}
which of the following has the function in the form y = f(u) and u = g(x)?
a. y = -21u, u = e^{x}
b. y = 21u, u = e^{-x}
c. y = -e^{u}, u = 21x
d. y = e^{u}, u = -21x
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Explanation:

Step1: Identify f(u) and g(x)

For the function $y = e^{-21x}$, if we let $u=-21x$ and $y = e^{u}$, it is in the form $y = f(u)$ and $u = g(x)$. So $f(u)=e^{u}$ and $g(x)=-21x$. This matches option D.

Step2: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$.
First, find $\frac{dy}{du}$. Since $y = e^{u}$, $\frac{dy}{du}=e^{u}$.
Second, find $\frac{du}{dx}$. Since $u=-21x$, $\frac{du}{dx}=-21$.
Then $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=e^{u}\cdot(-21)$.
Substitute $u = - 21x$ back into the expression, we get $\frac{dy}{dx}=-21e^{-21x}$.

Answer:

D. $y = e^{u},u=-21x$; $\frac{dy}{dx}=-21e^{-21x}$