Proof of Product & Quotient Rule

L3CALC

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Product Rule:

if $h(x)=f(x)g(x)$, then $h^{\prime }(x)=f^{\prime }(x)g(x)+g^{\prime }(x)f(x)$

Using first principles:

$$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)g(x+h)-f(x)g(x)}{h}$$

To evaluate $f(x+h)$ and $g(x+h)$, use first principles of $f^{\prime }(x)$ and $g^{\prime }(x)$:

$$f^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f(x+h)-f(x)}{h}$$

Then,

$$f(x+h)=h{f}^{\prime }(x)+f(x)$$ $$g(x+h)=h{g}^{\prime }(x)+g(x)$$

Hence,

$$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{[h{f}^{\prime }(x)+f(x)][h{g}^{\prime }(x)+g(x)]-f(x)g(x)}{h}$$ $$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{h^2{f}^{\prime }(x)g^{\prime }(x)+h{f}^{\prime }(x)g(x)+h{g}^{\prime }(x)f(x)+f(x)g(x)-f(x)g(x)}{h}$$ $$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{h(h{f}^{\prime }(x)g^{\prime }(x)+f^{\prime }(x)g(x)+g^{\prime }(x)f(x))}{h}$$ $$h^{\prime }(x)=\underset{h\to 0}{\lim }h{f}^{\prime }(x)g^{\prime }(x)+f^{\prime }(x)g(x)+g^{\prime }(x)f(x)$$ $$\boxed{h^{\prime }(x)=f^{\prime }(x)g(x)+g^{\prime }(x)f(x)}$$

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Quotient Rule:

if $h(x)=\frac{f(x)}{g(x)}$, then $h^{\prime }(x)=\frac{f^{\prime }(x)g(x)-g^{\prime }(x)f(x)}{g^2}$

$$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\left[\frac{f(x+h)}{g(x+h)}-\frac{f(x)}{g(x)}\right]}{h}$$

Using $f(x+h)$ and $g(x+h)$ from first principles:

$$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\left[\frac{h{f}^{\prime }(x)+f(x)}{h{g}^{\prime }(x)+g(x)}-\frac{f(x)}{g(x)}\right]}{h}$$ $$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\left[\frac{g(x)[h{f}^{\prime }(x)+f(x)]-f(x)[h{g}^{\prime }(x)+g(x)]}{g(x)[h{g}^{\prime }(x)+g(x)]}\right]}{h}$$ $$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\left[\frac{h{f}^{\prime }(x)f(x)+f(x)g(x)-hf(x)g^{\prime }(x)-f(x)g(x)}{hg(x)g^{\prime }(x)+(g(x){)}^2}\right]}{h}$$ $$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{\left[\frac{h{f}^{\prime }(x)f(x)-hf(x)g^{\prime }(x)}{hg(x)g^{\prime }(x)+(g(x){)}^2}\right]}{h}$$ $$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{h[f^{\prime }(x)g(x)-f(x)g^{\prime }(x)]}{h[hg(x)g^{\prime }(x)+(g(x){)}^2]}$$ $$h^{\prime }(x)=\underset{h\to 0}{\lim }\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{hg(x)g^{\prime }(x)+(g(x){)}^2}$$ $$\boxed{h^{\prime }(x)=\frac{f^{\prime }(x)g(x)-g^{\prime }(x)f(x)}{(g(x){)}^2}}$$