Calc2.61

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Let f(x) = a(x)\cdot b(x)

Then f(x+\Delta x) = a(x+\Delta x) \cdot b(x+\Delta x)

So \frac{ f(x+\Delta x) - f(x) }{\Delta x} = \frac{ a(x+\Delta x)b(x+\Delta x) - a(x)b(x) }{\Delta x} = \frac{ a(x+\Delta x)b(x+\Delta x) + a(x+\Delta x)b(x)- a(x+\Delta x)b(x) - a(x)b(x) }{\Delta x}

= \frac{ a(x+\Delta x)b(x+\Delta x) + a(x+\Delta x)b(x)}{\Delta x} - \frac{a(x+\Delta x)b(x) - a(x)b(x) }{\Delta x}

= a(x+\Delta x) \cdot \frac{b(x+\Delta x) +b(x)}{\Delta x} - \frac{a(x+\Delta x) +a(x)}{\Delta x} \cdot b(x)

Taking the limit: \lim_{\Delta x \to 0}\frac{ f(x+\Delta x) - f(x) }{\Delta x} = \lim_{\Delta x \to 0} \left [ a(x+\Delta x) \cdot \frac{b(x+\Delta x) +b(x)}{\Delta x} \right ]- \lim_{\Delta x \to 0} \left [\frac{a(x+\Delta x) +a(x)}{\Delta x} \cdot b(x) \right ]

Hence f'(x) = a(x) \cdot b'(x)+a'(x) \cdot b(x)


Note: It can be shown that \lim_{y \to c} g(y) \cdot h(y) = g(c)h(c) \, if \lim_{y \to c} g(y) = g(c) and \lim_{y \to c} h(y) = h(c)

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