for the following composite function, find an inner function ( u = g(x) ) and an outer function ( y = f(u) )…

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} ).\n\n( y=sqrt{7 x^{8}+6} )\n\nidentify the inner and outer functions. choose the correct answer.\n\na. ( u = g(x) = 7x^{8}+6 ) and ( y = f(u)=sqrt{u} )\nb. ( u = g(x) = 7x^{8}+6 ) and ( y = f(u)=u )\nc. ( u = g(x)=sqrt{x} ) and ( y = f(u)=7 u^{8}+6 )\nd. ( u = g(x)=x ) and ( y = f(u)=sqrt{u} )\n\n( \frac{dy}{dx}= )

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} ).\n\n( y=sqrt{7 x^{8}+6} )\n\nidentify the inner and outer functions. choose the correct answer.\n\na. ( u = g(x) = 7x^{8}+6 ) and ( y = f(u)=sqrt{u} )\nb. ( u = g(x) = 7x^{8}+6 ) and ( y = f(u)=u )\nc. ( u = g(x)=sqrt{x} ) and ( y = f(u)=7 u^{8}+6 )\nd. ( u = g(x)=x ) and ( y = f(u)=sqrt{u} )\n\n( \frac{dy}{dx}= )

Answer

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}})