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

write the function below in the form ( y = f(u) ) and ( u = g(x) ), then find ( \frac{dy}{dx} ) as a function of ( x ).\n( y=(5 x-12)^{5} )\nwrite ( y=(5 x-12)^{5} ) in the form ( y=f(u) ) and ( u=g(x) ). choose the correct functions ( f(u) ) and ( g(x) ) below.\n( \bigcirc mathrm{a} )\n( f(u)=u^{5} )\n( g(x)=5 x-12 )\n( \bigcirc mathrm{b} )\n( f(u)=5 u^{5} )\n( g(x)=x-12 )\n( \bigcirc mathrm{c} )\n( f(u)=(5 u-12)^{5} )\n( g(x)=5 x )\n( \bigcirc mathrm{d} )\n( f(u)=5 u-12 )\n( g(x)=x^{5} )

write the function below in the form ( y = f(u) ) and ( u = g(x) ), then find ( \frac{dy}{dx} ) as a function of ( x ).\n( y=(5 x-12)^{5} )\nwrite ( y=(5 x-12)^{5} ) in the form ( y=f(u) ) and ( u=g(x) ). choose the correct functions ( f(u) ) and ( g(x) ) below.\n( \bigcirc mathrm{a} )\n( f(u)=u^{5} )\n( g(x)=5 x-12 )\n( \bigcirc mathrm{b} )\n( f(u)=5 u^{5} )\n( g(x)=x-12 )\n( \bigcirc mathrm{c} )\n( f(u)=(5 u-12)^{5} )\n( g(x)=5 x )\n( \bigcirc mathrm{d} )\n( f(u)=5 u-12 )\n( g(x)=x^{5} )

Answer

Explanation:

Step1: Recall the chain - rule form

The chain - rule states that if (y = f(u)) and (u = g(x)), then (y=f(g(x))). We want to express (y=(5x - 12)^{5}) in the form (y = f(u)) and (u = g(x)). Let (u = g(x)=5x - 12) and (y=f(u)=u^{5}).

Step2: Use the chain - rule formula (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx})

First, find (\frac{dy}{du}): If (y = f(u)=u^{5}), then by the power rule (\frac{dy}{du}=\frac{d}{du}(u^{5})=5u^{4}). Second, find (\frac{du}{dx}): If (u = g(x)=5x - 12), then (\frac{du}{dx}=\frac{d}{dx}(5x - 12)=5).

Step3: Calculate (\frac{dy}{dx})

By the chain - rule (\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}). Substitute (u = 5x - 12), (\frac{dy}{du}=5u^{4}) and (\frac{du}{dx}=5) into the formula. (\frac{dy}{dx}=5u^{4}\cdot5). Replace (u) with (5x - 12), we get (\frac{dy}{dx}=25(5x - 12)^{4}).

Answer:

A. (f(u)=u^{5}), (g(x)=5x - 12) and (\frac{dy}{dx}=25(5x - 12)^{4})