1. **State the problem:** Given the function $y=3\sqrt{5x^2}$, find the derivative $y'$. 2. **Recall the formula:** The derivative of $y = a\sqrt{u}$ where $a$ is a constant and $u$ is a function of $x$ is given by $$y' = a \cdot \frac{1}{2\sqrt{u}} \cdot u'$$ 3. **Identify $u$ and $a$:** Here, $a=3$ and $u=5x^2$. 4. **Find $u'$:** $$u' = \frac{d}{dx}(5x^2) = 10x$$ 5. **Apply the derivative formula:** $$y' = 3 \cdot \frac{1}{2\sqrt{5x^2}} \cdot 10x = \frac{30x}{2\sqrt{5x^2}}$$ 6. **Simplify the expression:** $$y' = \frac{30x}{2\sqrt{5} |x|} = \frac{30x}{2\sqrt{5} x} = \frac{30}{2\sqrt{5}} = \frac{15}{\sqrt{5}}$$ Note: We used $|x|$ because $\sqrt{x^2} = |x|$. Assuming $x>0$, $|x|=x$. 7. **Rationalize the denominator:** $$y' = \frac{15}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} = \frac{15\sqrt{5}}{5} = 3\sqrt{5}$$ **Final answer:** $$y' = 3\sqrt{5}$$