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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]
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]
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