1. **State the problem:** We are given the function $y = x^2 e^{2x}$ and need to find the first derivative $y'$, the second derivative $y''$, and then calculate the price elasticity of demand $E = \frac{dy/dx \cdot x}{y}$. 2. **First derivative $y'$:** Use the product rule: If $y = u \cdot v$, then $y' = u'v + uv'$. Here, $u = x^2$ and $v = e^{2x}$. - Compute $u' = 2x$. - Compute $v' = \frac{d}{dx}e^{2x} = 2 e^{2x}$. - Apply product rule: $$y' = (2x)e^{2x} + x^2 (2 e^{2x}) = 2x e^{2x} + 2x^2 e^{2x} = 2x e^{2x}(1 + x).$$ 3. **Second derivative $y''$:** Differentiate $y' = 2x e^{2x}(1 + x)$. Again use product rule treating $w = 2x(1+x)$ and $z = e^{2x}$. - Simplify $w = 2x + 2x^2$. - Compute $w' = 2 + 4x$. - Compute $z' = 2 e^{2x}$. - Then: $$y'' = w' z + w z' = (2 + 4x)e^{2x} + (2x + 2x^2)(2 e^{2x}) = e^{2x} (2 + 4x) + 2 e^{2x}(2x + 2x^2).$$ - Factor $e^{2x}$ out: $$y'' = e^{2x} [2 + 4x + 4x + 4x^2] = e^{2x} (2 + 8x + 4x^2).$$ 4. **Price elasticity of demand $E$:** Defined as $$E = \frac{dy/dx \cdot x}{y} = \frac{y' x}{y}.$$ Plug in expressions: $$E = \frac{(2x e^{2x}(1+x)) x}{x^2 e^{2x}} = \frac{2x^2 e^{2x} (1+x)}{x^2 e^{2x}} = 2 (1 + x).$$ 5. **Final answers:** - First derivative: $$y' = 2x e^{2x}(1 + x).$$ - Second derivative: $$y'' = e^{2x}(2 + 8x + 4x^2).$$ - Price elasticity of demand: $$E = 2 (1 + x).$$