To solve for [tex]\(\frac{dy}{dx}\)[/tex] given the relationship [tex]\(x = 2 \tan y\)[/tex], we follow these steps:
1. Implicit Differentiation: Start by differentiating both sides of the equation [tex]\(x = 2 \tan y\)[/tex] with respect to [tex]\(x\)[/tex]. This allows us to find the derivative [tex]\(\frac{dy}{dx}\)[/tex].
[tex]\[
\frac{d}{dx}(x) = \frac{d}{dx}(2 \tan y)
\][/tex]
On the left side, the derivative of [tex]\(x\)[/tex] with respect to [tex]\(x\)[/tex] is 1:
[tex]\[
1 = \frac{d}{dx}(2 \tan y)
\][/tex]
2. Chain Rule Application: On the right side, we apply the chain rule where we differentiate [tex]\(2 \tan y\)[/tex] with respect to [tex]\(y\)[/tex], and then multiply by [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[
1 = 2 \sec^2 y \cdot \frac{dy}{dx}
\][/tex]
3. Solve for [tex]\(\frac{dy}{dx}\)[/tex]: Isolate [tex]\(\frac{dy}{dx}\)[/tex] by dividing both sides of the equation by [tex]\(2 \sec^2 y\)[/tex]:
[tex]\[
\frac{dy}{dx} = \frac{1}{2 \sec^2 y}
\][/tex]
4. Simplify using Trigonometric Identity: Recall the trigonometric identity [tex]\( \sec^2 y = 1 + \tan^2 y \)[/tex]. We substitute this into the equation:
[tex]\[
\frac{dy}{dx} = \frac{1}{2 (1 + \tan^2 y)}
\][/tex]
5. Express [tex]\(\tan y\)[/tex] in terms of [tex]\(x\)[/tex]: Given the initial equation [tex]\( x = 2 \tan y \)[/tex], we solve for [tex]\(\tan y\)[/tex]:
[tex]\[
\tan y = \frac{x}{2}
\][/tex]
6. Substitute [tex]\(\tan y\)[/tex] in the derivative: Replace [tex]\(\tan y\)[/tex] in the expression for [tex]\(\frac{dy}{dx}\)[/tex] with [tex]\(\frac{x}{2}\)[/tex]:
[tex]\[
\frac{dy}{dx} = \frac{1}{2 \left(1 + \left(\frac{x}{2}\right)^2 \right)}
\][/tex]
7. Simplify the expression: Simplify the denominator:
[tex]\[
\frac{dy}{dx} = \frac{1}{2 \left(1 + \frac{x^2}{4} \right)} = \frac{1}{2 \left( \frac{4 + x^2}{4} \right)} = \frac{1}{\frac{4 + x^2}{2}}
\][/tex]
Finally, by inverting the division in the fraction:
[tex]\[
\frac{dy}{dx} = \frac{2}{4 + x^2}
\][/tex]
This shows that the constant [tex]\(k\)[/tex] is 2, leading us to the final form of the derivative:
[tex]\[
\frac{dy}{dx} = \frac{2}{4 + x^2}
\][/tex]
