write the function in the form ( y = f(u) ) and ( u = g(x) ). then find ( \frac{dy}{dx} ) as a function of (…

write the function in the form ( y = f(u) ) and ( u = g(x) ). then find ( \frac{dy}{dx} ) as a function of ( x ).\n( y = e^{-8x} )\nwhich of the following has the function in the form ( y = f(u) ) and ( u = g(x) )?\na. ( y = -8u, u = e^{x} )\nb. ( y = -e^{u}, u = 8x )\nc. ( y = e^{u}, u = -8x )\nd. ( y = 8u, u = e^{-x} )

write the function in the form ( y = f(u) ) and ( u = g(x) ). then find ( \frac{dy}{dx} ) as a function of ( x ).\n( y = e^{-8x} )\nwhich of the following has the function in the form ( y = f(u) ) and ( u = g(x) )?\na. ( y = -8u, u = e^{x} )\nb. ( y = -e^{u}, u = 8x )\nc. ( y = e^{u}, u = -8x )\nd. ( y = 8u, u = e^{-x} )

Answer

Explanation:

Step1: Check each option

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