Implicit Differentiation wrt t.

Question

Suppose $x=x(t)$ and we want to differentiate wrt t. What would be the derivative of $(1-t)x^2=t^3$? Here is my try: $\frac{d}{dt} ((1-t)x^2=t^3) = (-x^2 + (1-t)\frac{d}{dt}x^2 = 3t^2) = (-x^2 + 2(1-t)x' = 3t^2)$. But on my answer sheet we have $-x^2 + (1-t)xx' = 3t^2$. Can someone explain the step where we get xx' instead of just x'.

Answer 1

By writing the derivative of $x^2$ as $2x$, you are computing $\frac {d}{dx} x^2$, not $\frac {d}{dt} x^2$.

Here, as $x$ is a function of one variable $t$,
$\frac {d}{dt} x^2 = \frac {dx}{dt} \frac {d}{dx} x^2 = 2 x x'$