Question

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

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 \(\frac{dy}{dx}=25(5x - 12)^{4}\)