Question

for the following composite function, find an inner function ( u = g(x) ) and an outer function ( y = f(u) ) such that ( y = f(g(x)) ). then calculate ( \frac{dy}{dx} ).

( y=sqrt{7 x^{8}+6} )

identify the inner and outer functions. choose the correct answer.

a. ( u = g(x) = 7x^{8}+6 ) and ( y = f(u)=sqrt{u} )
b. ( u = g(x) = 7x^{8}+6 ) and ( y = f(u)=u )
c. ( u = g(x)=sqrt{x} ) and ( y = f(u)=7 u^{8}+6 )
d. ( u = g(x)=x ) and ( y = f(u)=sqrt{u} )

( \frac{dy}{dx}= )

Explanation:

Step1: Identify inner and outer functions

For \(y = \sqrt{7x^{8}+6}\), the inner function \(u = g(x)\) is the expression inside the square - root. So \(u=g(x)=7x^{8}+6\). The outer function \(y = f(u)\) is the square - root function. So \(y = f(u)=\sqrt{u}=u^{\frac{1}{2}}\). So the correct choice is A.

Step2: Find the derivatives of \(f(u)\) and \(g(x)\)

• Derivative of \(y = f(u)\) with respect to \(u\):

Using the power rule \(\frac{d}{du}(u^{n})=nu^{n - 1}\), for \(y = u^{\frac{1}{2}}\), we have \(\frac{dy}{du}=\frac{1}{2}u^{\frac{1}{2}-1}=\frac{1}{2\sqrt{u}}\).

• Derivative of \(u = g(x)\) with respect to \(x\):

Using the power rule \(\frac{d}{dx}(ax^{n})=nax^{n - 1}\), for \(u = 7x^{8}+6\), we have \(\frac{du}{dx}=7\times8x^{8 - 1}+0 = 56x^{7}\).

Step3: Apply the chain rule

The chain rule states that \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\).
Substitute \(\frac{dy}{du}=\frac{1}{2\sqrt{u}}\) and \(\frac{du}{dx}=56x^{7}\) into the chain - rule formula. Since \(u = 7x^{8}+6\), we get:
\(\frac{dy}{dx}=\frac{1}{2\sqrt{7x^{8}+6}}\cdot56x^{7}\).
Simplify the expression: \(\frac{dy}{dx}=\frac{28x^{7}}{\sqrt{7x^{8}+6}}\).

Answer:

A. \(u = g(x)=7x^{8}+6\) and \(y = f(u)=\sqrt{u}\)

\(\frac{dy}{dx}=\frac{28x^{7}}{\sqrt{7x^{8}+6}}\)