1. **Problem:** Differentiate $y = \frac{3}{\sqrt{8x+3}}$ with respect to $x$. 2. **Formula and rules:** Recall that $\sqrt{u} = u^{1/2}$ and the derivative of $u^n$ with respect to $x$ is $n u^{n-1} \frac{du}{dx}$ (chain rule). 3. Rewrite the function: $$y = 3 (8x+3)^{-1/2}$$ 4. Differentiate using the chain rule: $$\frac{dy}{dx} = 3 \times \left(-\frac{1}{2}\right) (8x+3)^{-3/2} \times \frac{d}{dx}(8x+3)$$ 5. Compute the derivative inside: $$\frac{d}{dx}(8x+3) = 8$$ 6. Substitute back: $$\frac{dy}{dx} = 3 \times \left(-\frac{1}{2}\right) (8x+3)^{-3/2} \times 8$$ 7. Simplify constants: $$\frac{dy}{dx} = -12 (8x+3)^{-3/2}$$ 8. Final answer: $$\boxed{\frac{dy}{dx} = -\frac{12}{(8x+3)^{3/2}}}$$ This means the rate of change of $y$ with respect to $x$ is $-12$ divided by the quantity $(8x+3)$ raised to the power $3/2$.