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^{-19x} )
which of the following has the function in the form ( y = f(u) ) and ( u = g(x) )?
a. ( y = -19u, u = e^{x} )
b. ( y = -e^{u}, u = 19x )
c. ( y = 19u, u = e^{-x} )
d. ( y = e^{u}, u = -19x )

Explanation:

Step1: Check each option

For option D:
If \(y = e^{u}\) and \(u=-19x\), then substituting \(u\) into \(y\) gives \(y = e^{-19x}\), which is the original function.
For option A: If \(y=-19u\) and \(u = e^{x}\), then \(y=-19e^{x}
eq e^{-19x}\).
For option B: If \(y=-e^{u}\) and \(u = 19x\), then \(y=-e^{19x}
eq e^{-19x}\).
For option C: If \(y = 19u\) and \(u=e^{-x}\), then \(y = 19e^{-x}
eq e^{-19x}\).

Step2: Use the chain - rule

The chain - rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
If \(y = e^{u}\) and \(u=-19x\), then \(\frac{dy}{du}=e^{u}\) and \(\frac{du}{dx}=-19\).
Substitute \(u=-19x\) into \(\frac{dy}{du}\), we get \(\frac{dy}{du}=e^{-19x}\).
Then \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=e^{-19x}\cdot(-19)=-19e^{-19x}\).

Answer:

D. \(y = e^{u},u=-19x\) and \(\frac{dy}{dx}=-19e^{-19x}\)