Compute (d^2y)/(dx^2) for y=(2x+5)^11 . Show all work and your final answer using the html equation editor. Edit View Insert Format Tools Table square

To compute the second derivative of $y = (2x+5)^{11}$, we first need to find the first derivative.<br /><br />Let $u = 2x+5$. Then $y = u^{11}$.<br /><br />Using the chain rule, we have:<br /><br />$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$<br /><br />$\frac{dy}{du} = 11u^{10}$<br /><br />$\frac{du}{dx} = 2$<br /><br />So,<br /><br />$\frac{dy}{dx} = 11u^{10} \cdot 2 = 22u^{10}$<br /><br />Now, we need to find the second derivative. Again, using the chain rule:<br /><br />$\frac{d^{2}y}{dx^{2}} = \frac{d}{dx} \left( \frac{dy}{dx} \right)$<br /><br />$\frac{d^{2}y}{dx^{2}} = \frac{d}{dx} \left( 22u^{10} \right)$<br /><br />$\frac{d^{2}y}{dx^{2}} = 22 \cdot 10u^{9} \cdot \frac{du}{dx}$<br /><br />$\frac{d^{2}y}{dx^{2}} = 220u^{9} \cdot 2$<br /><br />$\frac{d^{2}y}{dx^{2}} = 440u^{9}$<br /><br />Finally, substituting $u = 2x+5$ back into the expression, we get:<br /><br />$\frac{d^{2}y}{dx^{2}} = 440(2x+5)^{9}$<br /><br />So, the second derivative of $y = (2x+5)^{11}$ is $\frac{d^{2}y}{dx^{2}} = 440(2x+5)^{9}$.