Multiplication is repeated addition: $$x^2 = \underbrace{x+\cdots+x}_{x\text{ times}}$$
$$\frac{d}{dx} \big(x^2\big)\qquad\qquad$$
$$\qquad= {\underbrace{1+\cdots+1}_{x\text{ times}}}$$
$$\qquad+ \underbrace{x+\cdots+x}_{1\text{ times}}$$
$$\qquad= x + x = 2x .$$