Question

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 = -$e^{u}$, u = 21x
c. y = $e^{u}$, u = -21x
d. y = 21u, u = $e^{-x}$

Explanation:

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

Given $y = e^{-21x}$, we can set $y=f(u)=e^{u}$ and $u = g(x)=-21x$. So the correct form is C.

Step2: Apply 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 $u=-21x$, $\frac{du}{dx}=-21$.

Step3: Calculate $\frac{dy}{dx}$

Substitute $u = - 21x$ and the values of $\frac{dy}{du}$ and $\frac{du}{dx}$ into the chain - rule formula. $\frac{dy}{dx}=e^{u}\cdot(-21)$. Replace $u$ with $-21x$, we get $\frac{dy}{dx}=-21e^{-21x}$.

Answer:

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