The Product Rule (geometrically)

Theorem (The Product Rule): 

If $f$ and $g$ are differentiable functions, then the derivative of the product $fg$ is

 $ddx[f(x)g(x)]=f(x)\cdot ddxg(x)+g(x)\cdot ddxf(x)$. 

We will now derive the product rule geometrically. 

Suppose $f$ and $g$ are positive functions that are differentiable, and denote $\Delta x$ to be the change in $f(x)$ and $g(x)$. Therefore, we can denote the change in $f(x)$ and the change in $g(x)$ to be: 

(1) $\Delta f(x)=f(x+\Delta x)-f(x)\Delta g(x)=g(x+\Delta x)-g(x)$ 

The diagram above illustrates the area of $f(x)g(x)$ as well as the area 

$(f(x)+\Delta f(x))(g(x)+\Delta g(x))$.

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