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^{-8x} )
which of the following has the function in the form ( y = f(u) ) and ( u = g(x) )?
a. ( y = -8u, u = e^{x} )
b. ( y = -e^{u}, u = 8x )
c. ( y = e^{u}, u = -8x )
d. ( y = 8u, u = e^{-x} )

Explanation:

Step1: Check each option

• For option A: If \(y = - 8u\) and \(u=e^{x}\), then \(y=-8e^{x}

eq e^{-8x}\).

• For option B: If \(y=-e^{u}\) and \(u = 8x\), then \(y=-e^{8x}

eq e^{-8x}\).

• For option C: If \(y = e^{u}\) and \(u=-8x\), then \(y = e^{-8x}\).
• For option D: If \(y = 8u\) and \(u=e^{-x}\), then \(y = 8e^{-x}

eq e^{-8x}\).

Step2: Use the chain - rule

The chain - rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
Since \(y = e^{u}\) and \(u=-8x\), \(\frac{dy}{du}=e^{u}\) and \(\frac{du}{dx}=-8\).
Substitute \(u = - 8x\) into \(\frac{dy}{du}\), we get \(\frac{dy}{du}=e^{-8x}\).
Then \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=e^{-8x}\cdot(-8)=-8e^{-8x}\)

Answer:

C. \(y = e^{u},u=-8x\) and \(\frac{dy}{dx}=-8e^{-8x}\)