Question

Differentiate. y = 19x + 6)7-6 719x + 6)6 2V19x + 637 - 6 6319x + 66 (x + 6) = 6 6319x + 6)6 2x + 677-6 719x + 6)6 10x + 6) - 6

Answer 1

We will investigate the process of differentiation.

The process of differentiation constitutes a multiplication of smaller differential processes as follows:

`\text{Power}\cdot\text{Function ... Rule}`

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:

`y\text{ = }\sqrt[]{(9x+6)^7-6}`

Now we will break down the above parent function into power and a function ( f ( x ) ) which is raised to a power as follows:

`y=(f(x))^{(1)/(2)}^{}`

Where,

`f(x)=(9x+6)^7\text{ - 6}`

Now we can write the differential of the parent function ( y ) by applying the power differential rule as follows:

`\begin{gathered} y^(\prime)\text{ = }(1)/(2)\cdot(f(x))^{(1)/(2)-1}\cdot(f^(\prime)(x)) \\ \\ \textcolor{#FF7968}{y^(\prime)}\text{\textcolor{#FF7968}{ = }}\textcolor{#FF7968}{(f^(\prime)(x))/(2\cdot f(x))} \end{gathered}`

Now we will differentiate the function f ( x ). However, the function f ( x ) can be broken downto another power-function formulation as follows:

`f(x)=(g(x))^7\text{ - 6}`

Where,

`g\text{ ( x ) = 9x + 6}`

Now again apply the power-function rule and evaluate f ' ( x ) as follows:

`\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}`

Now we plug in the result of f ' ( x ) in the first power-function differentiation as follows:

`\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}}`