To find [tex]\(\frac{dy}{dx}\)[/tex] for the equation [tex]\(x^2 + 2m = y^2\)[/tex], we need to use implicit differentiation. Let's go through the steps together:
1. Given Equation:
[tex]\[
x^2 + 2m = y^2
\][/tex]
2. Differentiate both sides with respect to [tex]\(x\)[/tex]:
Since [tex]\(m\)[/tex] is a constant, its derivative with respect to [tex]\(x\)[/tex] is [tex]\(0\)[/tex]. Differentiating the equation term by term:
[tex]\[
\frac{d}{dx}(x^2) + \frac{d}{dx}(2m) = \frac{d}{dx}(y^2)
\][/tex]
[tex]\[
\frac{d}{dx}(x^2) + \frac{d}{dx}(2m) = \frac{d}{dx}(y^2)
\][/tex]
3. Compute the derivatives:
[tex]\[
\frac{d}{dx}(x^2) = 2x
\][/tex]
[tex]\[
\frac{d}{dx}(2m) = 0 \quad (\text{since } m \text{ is a constant})
\][/tex]
[tex]\[
\frac{d}{dx}(y^2) = 2y \frac{dy}{dx} \quad (\text{by the chain rule})
\][/tex]
4. Substitute these results back into the differentiated equation:
[tex]\[
2x + 0 = 2y \frac{dy}{dx}
\][/tex]
5. Simplify the equation:
[tex]\[
2x = 2y \frac{dy}{dx}
\][/tex]
6. Solve for [tex]\(\frac{dy}{dx}\)[/tex]:
[tex]\[
\frac{dy}{dx} = \frac{2x}{2y}
\][/tex]
[tex]\[
\frac{dy}{dx} = \frac{x}{y}
\][/tex]
Therefore, the derivative [tex]\(\frac{dy}{dx}\)[/tex] is:
[tex]\(\boxed{\frac{x}{y}}\)[/tex]
None of the provided answer choices match this derivative directly, but based on the given work, the correct [tex]\(\frac{dy}{dx}\)[/tex] should be understood as [tex]\(\frac{x}{y}\)[/tex]. The steps and results are verified and correct as per conventional calculus methods.
