Question

Please help me work through this I am just a little confused

Answer 1

Step 1:

In this question, we are given the following:

Step 2:

The details of the solution are as follows:

`\begin{gathered} Given\text{ the function: y =6 + }(6)/(x)+(3)/(x^2) \\ Differentiating\text{ the function wwith respect to x, we have that:} \\ (dy)/(dx)=\text{ }(-6)/(x^2)-(6)/(x^3) \end{gathered}`

`\begin{gathered} (dy)/(dx)=\text{ }(-6)/(x^2)-(6)/(x^3) \\ put\text{ x = 3, we have that:} \end{gathered}`
`(dy)/(dx)=\text{ m = }(-6)/(3^2)-(6)/(3^3)=(-6)/(9)-(6)/(27)=(-18-6)/(27)=(-24)/(27)=(-8)/(9)`
`\begin{gathered} put\text{ the value of x = 3 in } \\ \text{y = 6 +}(6)/(x)+\text{ }(3)/(x^2)\text{, we have that:} \\ y_1\text{= 6 + }(6)/(3)+(3)/(3^2^)=6\text{ + 2 + }(1)/(3)=\text{ 8 }(1)/(3)=(25)/(3) \end{gathered}`

Now, we have that:

`\begin{gathered} (x_1,\text{ y}_1)\text{ = \lparen 3, }(25)/(3)) \\ and\text{ m = }(-8)/(9) \end{gathered}`
`\begin{gathered} Using\text{ y -y}_1=\text{ m \lparen x - x}_1) \\ y\text{ -}(25)/(3)=(-8)/(9)(\text{ x - 3\rparen} \end{gathered}`
`\begin{gathered} Multiplying\text{ through by 9, we have that:} \\ 9\text{y -75 = - 8 \lparen x- 3\rparen} \\ 9y\text{ -75 = -8x + 24} \\ 9y\text{ + 8x = 24 + 75} \\ \text{8x + 9y = 99} \end{gathered}`

The final answer is:

`8x+\text{ 9y = 99}`