The question:

Let $\displaystyle z = \sqrt{x^2 + y^2}, x = t^2, y = t^3$. Use a chain rule to ﬁnd dz/dt.

My attempt:

$\displaystyle \frac{dz}{dt} = \frac{\partial{z}}{\partial{x}}\frac{dx}{dt} + \frac{\partial{z}}{\partial{y}}\frac{dy}{dt}$

$\displaystyle \frac{\partial{z}}{\partial{x}} = \frac{x}{\sqrt{x^2 + y^2}}$

$\displaystyle \frac{\partial{z}}{\partial{y}} = \frac{y}{\sqrt{x^2 + y^2}}$

$\displaystyle \frac{dx}{dt} = 2t$

$\displaystyle \frac{dy}{dt} = 3t^2$

So we get:

$\displaystyle \frac{t^2}{\sqrt{t^4 + t^6}}(2t) + \frac{t^3}{\sqrt{t^4 + t^6}}(3t^2)$

= $\displaystyle \frac{t^2(2t + 3t^3)}{t^2\sqrt{1 + t^2}}$

= $\displaystyle \frac{2t + 3t^3}{\sqrt{1 + t^2}}$

However, the answer is apparently:

$\displaystyle \frac{2t^2 + 3t^4}{\sqrt{1 + t^2}}$

What have I done incorrectly? Thanks.