Multiplication is repeated addition: $$ x^2 = \underbrace{x+\cdots+x}_{x\text{ times}} $$
Differentiate (remember the chain rule for partial derivatives):
$$\frac{d}{dx} \big(x^2\big)\qquad\qquad$$
$$ \qquad= {\underbrace{1+\cdots+1}_{x\text{ times}}}$$
$$\qquad+ \underbrace{x+\cdots+x}_{1\text{ times}}$$
$$ \qquad= x + x = 2x .$$
