Question

Don’t get part b of the question. Very confusing any chance you may help me with this please.

Answer 1

To solve this problem, first, we will solve the given equation for y:

`\begin{gathered} x=3\tan 2y, \\ \tan 2y=(x)/(3), \\ 2y=\arctan ((x)/(3)), \\ y=(\arctan((x)/(3)))/(2)=(1)/(2)\arctan ((x)/(3))\text{.} \end{gathered}`

Once we have the above equation, now we compute the derivative. To compute the derivative we will use the following properties of derivatives:

`\begin{gathered} (d)/(dx)\arctan (x)=(1)/(x^2+1), \\ (dkf(x))/(dx)=k(df(x))/(dx). \end{gathered}`

Where k is a constant.

First, we use the second property above, and get that:

`(d(\arctan((x)/(3)))/(2))/(dx)=(d\arctan ((x)/(3))*(1)/(2))/(dx)=(1)/(2)(d\arctan ((x)/(3)))/(dx)\text{.}`

Now, from the chain rule, we get:

`(dy)/(dx)=(1)/(2)\frac{d\text{ arctan(}(x)/(3))}{dx}=(1)/(2)(d\arctan ((x)/(3)))/(dx)|_{(x)/(3)}(d(x)/(3))/(dx)\text{.}`

Finally, computing the above derivatives (using the rule for the arctan), we get:

`(dy)/(dx)=(1)/(2)((1)/(3))/((x^2)/(9)+1)=(1)/(6)((1)/((x^2)/(9)+1))=(3)/(2(x^2+9)).`

Answer:

`(3)/(2(x^2+9)).`