We will investigate the process of differentiation.
The process of differentiation constitutes a multiplication of smaller differential processes as follows:
[tex]\text{Power}\cdot\text{Function ... Rule}[/tex]The power-function rule of differentiation involves two sub-processes of differentiating the power multiplied by the differential of function within.
We are given the following parent function as follows:
[tex]y\text{ = }\sqrt[]{(9x+6)^7-6}[/tex]Now we will break down the above parent function into power and a function ( f ( x ) ) which is raised to a power as follows:
[tex]y=(f(x))^{\frac{1}{2}}^{}[/tex]Where,
[tex]f(x)=(9x+6)^7\text{ - 6}[/tex]Now we can write the differential of the parent function ( y ) by applying the power differential rule as follows:
[tex]\begin{gathered} y^{\prime}\text{ = }\frac{1}{2}\cdot(f(x))^{\frac{1}{2}-1}\cdot(f^{\prime}(x)) \\ \\ \textcolor{#FF7968}{y^{\prime}}\text{\textcolor{#FF7968}{ = }}\textcolor{#FF7968}{\frac{f^{\prime}(x)}{2\cdot f(x)}} \end{gathered}[/tex]Now we will differentiate the function f ( x ). However, the function f ( x ) can be broken downto another power-function formulation as follows:
[tex]f(x)=(g(x))^7\text{ - 6}[/tex]Where,
[tex]g\text{ ( x ) = 9x + 6}[/tex]Now again apply the power-function rule and evaluate f ' ( x ) as follows:
[tex]\begin{gathered} f^{\prime}(x)\text{ = 7}\cdot(g(x))^{7-1}\cdot g^{\prime}(x) \\ f^{\prime}(x)\text{ = 7}\cdot(9x+6)^6\cdot9 \\ \textcolor{#FF7968}{f^{\prime}(x)}\text{\textcolor{#FF7968}{ = 63}}\textcolor{#FF7968}{\cdot(9x+6)^6} \end{gathered}[/tex]Now we plug in the result of f ' ( x ) in the first power-function differentiation as follows:
[tex]\textcolor{#FF7968}{y}^{\textcolor{#FF7968}{\prime}}\text{\textcolor{#FF7968}{ = }}\textcolor{#FF7968}{\frac{63\cdot(9x+6)^6}{2\sqrt[]{(9x+6)^7-6}},,,}\text{\textcolor{#FF7968}{ option C}}[/tex]