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3.6 Leibniz’s rules for derivatives

Theorem

Let $f\,$ and $g\,$ be differentiable at $a\,$ , and $C$ be a constant. Then the sum and product of these functions are differentiable at $a\,$ with:

(i)\quad\bigl(Cf\bigr)^{\prime}(a)=Cf^{\prime}(a)

(ii)\quad\bigl(f+g\bigr)^{\prime}(a)=f^{\prime}(a)+g^{\prime}(a)

(iii)\quad(fg)^{\prime}(a)=f^{\prime}(a)g(a)+f(a)g^{\prime}(a)

(iv)\quad\Bigl({{f}\over{g}}\Bigr)^{\prime}(a)={{f^{\prime}(a)g(a)-f(a)g^{% \prime}(a)}\over{g(a)^{2}}}\qquad(g(a)\neq 0).

If you prefer Leibniz’s notation, then see 3.30.