1. The problem asks to find the derivative $\frac{dy}{dx}$ if $y = \sin^{-1}(5x)$.\n\n2. Recall the derivative formula for the inverse sine function: $$\frac{d}{dx} \sin^{-1}(u) = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}$$ where $u$ is a function of $x$.\n\n3. Here, $u = 5x$, so $\frac{du}{dx} = 5$.\n\n4. Substitute $u$ and $\frac{du}{dx}$ into the formula: $$\frac{dy}{dx} = \frac{1}{\sqrt{1-(5x)^2}} \cdot 5 = \frac{5}{\sqrt{1-25x^2}}$$\n\n5. This matches option C except for the negative sign. The derivative of $\sin^{-1}(x)$ is positive, so the negative sign in option C is incorrect.\n\n6. Therefore, the correct derivative is $$\frac{dy}{dx} = \frac{5}{\sqrt{1-25x^2}}$$ which corresponds to none of the options exactly, but closest to option C without the negative sign. Since the problem's options include a negative sign in C, the correct choice is option B: $$\frac{5}{1+25x^2}$$ is incorrect because the denominator should be a square root and subtraction, not addition. Option D is missing the factor 5. Option A is incorrect for the same reason as B.\n\n7. The correct derivative is $$\frac{dy}{dx} = \frac{5}{\sqrt{1-25x^2}}$$ which is not exactly listed but closest to option C ignoring the negative sign.\n\nFinal answer: $$\frac{dy}{dx} = \frac{5}{\sqrt{1-25x^2}}$$