Differentiation Rules

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本文介绍了微积分中的三个基本求导法则：和差法则、乘积法则和商法则，并给出了详细的数学证明过程。

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Differentiation Rules

1. The Sum Rule

In calculus, the sum rule in differentiation is a method of finding the derivative of a function that is the sum of two other functions for which derivatives exist.[1]

Given: $h\left( x \right) = f\left( x \right) + g\left( x \right)$

Proofs: $h'\left( x \right) = f'\left( x \right) + g'\left( x \right)$

h' (x) = lim Δ x \to 0 h ( x + Δ x ) - h ( x ) Δ x = lim Δ x \to 0 f ( x + Δ x ) + g ( x + Δ x ) - f ( x ) - g ( x ) Δ x = lim Δ x \to 0 f ( x + Δ x ) - f ( x ) + g ( x + Δ x ) - g ( x ) Δ x = lim Δ x \to 0 f ( x + Δ x ) - f ( x ) Δ x + g ( x + Δ x ) - g ( x ) Δ x = f' (x) + g' (x)

2. The Product Rule

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions.[2]

Given: $h\left( x \right) = f\left( x \right) \cdot g\left( x \right)$

Proofs: $h'\left( x \right) = f'\left( x \right)g\left( x \right) + f\left( x \right)g'\left( x \right)$

h' (x) = lim Δ x \to 0 h ( x + Δ x ) - h ( x ) Δ x = lim Δ x \to 0 f ( x + Δ x ) \cdot g ( x + Δ x ) - f ( x ) \cdot g ( x ) Δ x = lim Δ x \to 0 f ( x + Δ x ) \cdot g ( x + Δ x ) - [ f ( x ) \cdot g ( x + Δ x ) + f ( x ) \cdot g ( x + Δ x ) ] - f ( x ) \cdot g ( x ) Δ x = lim Δ x \to 0 [ f ( x + Δ x ) - f ( x ) ] \cdot g ( x + Δ x ) + f ( x ) \cdot [ g ( x + Δ x ) - g ( x ) ] Δ x = lim Δ x \to 0 f ( x + Δ x ) - f ( x ) Δ x \cdot lim Δ x \to 0 g (x + Δ x) + lim Δ x \to 0 f (x + Δ x) \cdot lim Δ x \to 0 g ( x + Δ x ) - g ( x ) Δ x = f' (x) g (x) + f (x) g' (x)

$\begin{align} h'\left( x \right) &= \lim\limits_{\Delta x \rightarrow0} \frac{ h\left(x + \Delta x\right) - h \left( x \right)}{\Delta x} \\ &= \lim\limits_{\Delta x \rightarrow0} \frac{ f\left(x + \Delta x\right) \cdot g\left(x + \Delta x\right) - f \left( x \right) \cdot g \left( x \right)}{\Delta x} \\ &= \lim\limits_{\Delta x \rightarrow0} \frac{ f\left(x + \Delta x\right) \cdot g\left(x + \Delta x\right) - \left[ f\left(x \right) \cdot g\left(x + \Delta x \right) + f\left(x \right) \cdot g\left(x + \Delta x \right)\right] - f \left( x \right) \cdot g \left( x \right)}{\Delta x} \\ &= \lim\limits_{\Delta x \rightarrow0} \frac{ \left[f\left(x + \Delta x\right) -f\left( x \right)\right] \cdot g\left(x + \Delta x\right) + f\left( x \right) \cdot \left[g\left(x + \Delta x\right) -g\left( x \right)\right] }{\Delta x} \\ &= \lim\limits_{\Delta x \rightarrow0} \frac{ f\left(x + \Delta x\right) -f\left( x \right)}{\Delta x} \cdot \lim\limits_{\Delta x \rightarrow0} g\left(x + \Delta x\right) + \lim\limits_{\Delta x \rightarrow0} f\left(x + \Delta x\right) \cdot \lim\limits_{\Delta x \rightarrow0} \frac{ g\left(x + \Delta x\right) -g\left( x \right)}{\Delta x} \\ &= f'\left( x \right)g\left( x \right) + f\left( x \right)g'\left( x \right) \end{align}$

3. The Quotient Rule

In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.[3]

Given: $h\left( x \right) = \frac {f\left( x \right)} {g\left( x \right)}$

Proofs: $h'\left( x \right) = \frac{f'\left( x \right)g\left( x \right) -f\left( x \right) g'\left( x \right)}{g\left( x \right)^2}$

h' (x) = lim Δ x \to 0 h ( x + Δ x ) - h ( x ) Δ x = lim Δ x \to 0 f ( x + Δ x ) g ( x + Δ x ) - f ( x ) g ( x ) Δ x = lim Δ x \to 0 f ( x + Δ x ) g ( x ) - f ( x ) g ( x + Δ x ) Δ x \cdot g ( x ) g ( x + Δ x ) = lim Δ x \to 0 f ( x + Δ x ) g ( x ) - f ( x ) g ( x + Δ x ) Δ x \cdot 1 g ( x ) g ( x + Δ x ) = [lim Δ x \to 0 f ( x + Δ x ) - f ( x ) Δ x \cdot lim Δ x \to 0 g (x) - lim Δ x \to 0 f (x) \cdot lim Δ x \to 0 g ( x + Δ x ) - g ( x ) Δ x] \cdot 1 g ( x ) 2 = f ' ( x ) g ( x ) - f ( x ) g ' ( x ) g ( x ) 2

$\begin{align} h'\left( x \right) &= \lim\limits_{\Delta x \rightarrow0} \frac{ h\left(x + \Delta x\right) - h \left( x \right)}{\Delta x} \\ &= \lim\limits_{\Delta x \rightarrow0} \frac{\frac {f\left( x + \Delta x \right)} {g\left( x + \Delta x \right)} - \frac{f\left( x \right)}{g\left( x \right)}}{\Delta x} \\ &= \lim\limits_{\Delta x \rightarrow0} \frac{f\left( x + \Delta x \right) g\left( x \right) - f\left( x \right)g\left( x + \Delta x \right)}{\Delta x \cdot g\left( x \right)g\left( x + \Delta x \right)}\\ &= \lim\limits_{\Delta x \rightarrow0} \frac{f\left( x + \Delta x \right) g\left( x \right) - f\left( x \right)g\left( x + \Delta x \right)}{\Delta x} \cdot \frac{1}{g\left( x \right)g\left( x + \Delta x \right)}\\ &= \left[\lim\limits_{\Delta x \rightarrow0} \frac{ f\left(x + \Delta x\right) -f\left( x \right)}{\Delta x} \cdot \lim\limits_{\Delta x \rightarrow0} g\left(x\right) - \lim\limits_{\Delta x \rightarrow0} f\left(x \right) \cdot \lim\limits_{\Delta x \rightarrow0} \frac{ g\left(x + \Delta x\right) -g\left( x \right)}{\Delta x}\right] \cdot \frac{1}{g\left( x \right)^2}\\ &= \frac{f'\left( x \right)g\left( x \right) -f\left( x \right) g'\left( x \right)}{g\left( x \right)^2} \end{align}$