The degree of the given equation is $\frac{dy}{dx}-x={\left(y-x\frac{dy}{dx}\right)}^{-6}$

Given: $\frac{dy}{dx}-x=\frac{1}{(y-x\frac{dy}{dx}{)}^6}$ To find the degree, we must express the differential equation in a form free from radicals and fractions regarding the derivatives.Cross-multiplying:
$$\left(\frac{dy}{dx}-x\right){\left(y-x\frac{dy}{dx}\right)}^6=1$$
Expanding the term $(y-x\frac{dy}{dx}{)}^6$ using binomial expansion, the highest power of the derivative $\frac{dy}{dx}$ within that term is $(\frac{dy}{dx}{)}^6$.When multiplied by the first $\frac{dy}{dx}$ term, the highest power becomes:
$${\left(\frac{dy}{dx}\right)}^1\cdot {\left(\frac{dy}{dx}\right)}^6={\left(\frac{dy}{dx}\right)}^7$$
Thus, the degree of the differential equation is $7$.