1. **Problem Statement:** Find the derivative with respect to $x$ of the integral $$\frac{d}{dx} \left( \int_{x^2}^x t^2 \, dt \right).$$ 2. **Recall the Fundamental Theorem of Calculus (FTC):** - If $F(x) = \int_a^x f(t) \, dt$, then $$\frac{d}{dx} F(x) = f(x).$$ - For variable limits, $$\frac{d}{dx} \int_{u(x)}^{v(x)} f(t) \, dt = f(v(x)) \cdot v'(x) - f(u(x)) \cdot u'(x).$$ 3. **Apply FTC to the problem:** - Here, $u(x) = x^2$, $v(x) = x$, and $f(t) = t^2$. - Compute derivatives: $$u'(x) = 2x, \quad v'(x) = 1.$$ 4. **Evaluate the derivative:** $$\frac{d}{dx} \int_{x^2}^x t^2 \, dt = f(v(x)) \cdot v'(x) - f(u(x)) \cdot u'(x) = (x)^2 \cdot 1 - (x^2)^2 \cdot 2x = x^2 - (x^4) \cdot 2x = x^2 - 2x^5.$$ 5. **Final answer:** $$\boxed{\frac{d}{dx} \int_{x^2}^x t^2 \, dt = x^2 - 2x^5}.$$