If $u = x^2 + y^2$, where $x = t^2$ and $y = 2t$, the value of the total derivative $\frac{du}{dt}$ evaluated at $t = 1$ is ________ (answer in integer).
Answer Key:
12
Step-by-Step Solution & Explanation
By the chain rule for total differentiation:
$\frac{du}{dt} = \frac{\partial u}{\partial x} \frac{dx}{dt} + \frac{\partial u}{\partial y} \frac{dy}{dt}$
Computing individual terms:
$\frac{\partial u}{\partial x} = 2x, \quad \frac{dx}{dt} = 2t$
$\frac{\partial u}{\partial y} = 2y, \quad \frac{dy}{dt} = 2$
At $t = 1$, we have $x = 1^2 = 1$ and $y = 2(1) = 2$:
$\frac{du}{dt} = 2(1)(2(1)) + 2(2)(2) = 4 + 8 = 12$