1. **State the problem:** Differentiate the function $y = \sqrt[7]{x^3}$ with respect to $x$. 2. **Rewrite the function using exponents:** Recall that the seventh root can be written as a fractional exponent: $$y = x^{\frac{3}{7}}$$ 3. **Use the power rule for differentiation:** The power rule states that if $y = x^n$, then $$\frac{dy}{dx} = n x^{n-1}$$ 4. **Apply the power rule:** Here, $n = \frac{3}{7}$, so $$\frac{dy}{dx} = \frac{3}{7} x^{\frac{3}{7} - 1} = \frac{3}{7} x^{-\frac{4}{7}}$$ 5. **Simplify the expression:** Negative exponents can be rewritten as reciprocals: $$\frac{dy}{dx} = \frac{3}{7} \frac{1}{x^{\frac{4}{7}}} = \frac{3}{7 x^{\frac{4}{7}}}$$ **Final answer:** $$\frac{dy}{dx} = \frac{3}{7 x^{\frac{4}{7}}}$$