Leibniz product rule

Leibniz product rule

Also known as Leibniz rule or product rule is a formula used to find the derivatives of products of two or more functions. In Leibniz's notation it's written as \[\frac{d}{dx}(u\cdot v)=\frac{du}{dx}\cdot v+u\cdot \frac{dv}{dx}\], or in Lagrange's notation it's written as \[(u\cdot v)^{\prime}=u^{\prime}\cdot v+u\cdot v^{\prime}\].

Proof

Let \[h(x)=f(x)\cdot g(x)\]:

\begin{align*} h^{\prime}(x)&=\lim_{\Delta x\to0}\frac{h(x+\Delta x)-h(x)}{\Delta x}\\ &=\lim_{\Delta x\to0}\frac{f(x+\Delta x)\cdot g(x+\Delta x)-f(x)\cdot g(x)}{\Delta x}\\ &=\lim_{\Delta x\to0}\frac{f(x+\Delta x)\cdot g(x+\Delta x)-f(x)\cdot g(x+\Delta x)+f(x)\cdot g(x+\Delta x)-f(x)\cdot g(x)}{\Delta x}\\ &=\lim_{\Delta x\to0}\frac{g(x+\Delta x)(f(x+\Delta x)-f(x))+f(x)(g+(x+\Delta x)-g(x))}{\Delta x}\\ &=\lim_{\Delta x\to0}g(x+\Delta x)\cdot \lim_{\Delta x\to0}\frac{f(x+\Delta x)-f(x)}{\Delta x}+\lim_{\Delta x\to0}f(x)\cdot \lim_{\Delta x\to0}\frac{g(x+\Delta x)-g(x)}{\Delta x}\\ &=g(x)\cdot f^{\prime}(x)+f(x)\cdot g^{\prime}(x)\\ \end{align*}
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