Derivative of a quadratic form with respect to a scalar

Question

How would I take a derivative of a quadratic form with respect to a scalar, i.e.
$$\frac{dx(t)^T\mathbf{Y}x(t)}{dt}$$
I have already tried splitting the quadratic form into its elements, and also applied the product rule, but it did not help. Any help is appreciated.

Answer 1

The hint of CaptainLama is correct.

An alternative way is by expanding the form and observing what square terms

$$a\,x_0(t)^2\to 2a\,x_0(t)x_0'(t)$$

and double product terms

$$2b\,x_0(t)x_1(t)\to 2b\,x_0(t)'x_1(t)+2b\,x_0(t)x_1'(t)$$

become.

This should convince you that the derivative is the bilinear form

$$2x'(t)^TYx(t).$$