Let $I\subseteq\mathbb{R}$ be an interval and let $f,g:I\to\mathbb{R}$ be differentiable at a point $x\in I^\circ$ (interior of $I$). Let $c\in\mathbb{R}$.

Proposition (standard derivative rules):

• Linearity: $(f+g)'(x)=f'(x)+g'(x),\qquad (cf)'(x)=c f'(x).$
• Product rule: $(fg)'(x)=f'(x)g(x)+f(x)g'(x).$
• Quotient rule: if $g(x)\neq 0$, then $\left(\frac{f}{g}\right)'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}.$
• Chain rule : if $g$ is differentiable at $x$ and $f$ is differentiable at $g(x)$, then $(f\circ g)'(x)=f'(g(x))\,g'(x).$

These formulas are the computational backbone of differential calculus; they are proved directly from the limit definition of the derivative .

Proof sketch: Use algebraic manipulation of difference quotients, e.g. $ \frac{f(x+h)g(x+h)-f(x)g(x)}{h}

\frac{f(x+h)-f(x)}{h}g(x+h)+f(x)\frac{g(x+h)-g(x)}{h}, $ then pass to the limit using continuity of differentiable functions.