5. Find (dy)/(dx) if y=3z^2-5z+2 where z=x^2-1 and x=1

To find $\frac{dy}{dx}$, we need to use the chain rule. The chain rule states that if $y$ is a function of $z$ and $z$ is a function of $x$, then $\frac{dy}{dx} = \frac{dy}{dz} \cdot \frac{dz}{dx}$.<br /><br />Given $y = 3z^2 - 5z + 2$ and $z = x^2 - 1$, we can find $\frac{dy}{dz}$ and $\frac{dz}{dx}$.<br /><br />First, let's find $\frac{dy}{dz}$:<br />$\frac{dy}{dz} = \frac{d}{dz}(3z^2 - 5z + 2) = 6z - 5$<br /><br />Next, let's find $\frac{dz}{dx}$:<br />$\frac{dz}{dx} = \frac{d}{dx}(x^2 - 1) = 2x$<br /><br />Now, we can substitute these values into the chain rule formula:<br />$\frac{dy}{dx} = \frac{dy}{dz} \cdot \frac{dz}{dx} = (6z - 5) \cdot 2x$<br /><br />Finally, substitute $z = x^2 - 1$ into the expression:<br />$\frac{dy}{dx} = (6(x^2 - 1) - 5) \cdot 2x = (6x^2 - 6 - 5) \cdot 2x = (6x^2 - 11) \cdot 2x = 12x^3 - 22x$<br /><br />Therefore, the correct answer is $\frac{dy}{dx} = 12x^3 - 22x$.