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

Don’t get part b of the question. Very confusing any chance you may help me with this-example-1
User Diego Palomar
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1 Answer

21 votes
21 votes

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)).

User Wauna
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