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ACMMM106

4.3 Product and Quotient Rules

The Product Rule

  • Assume that there are functions f(x),\ g(x),\ h(x) where h(x)=f(x)\cdot g(x). Hence,

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

  • Using Leibniz Notation, it is equivalent by saying

\frac{d y}{d x}=u \frac{d v}{d x}+v \frac{d u}{d x}

where u=g(x), v=f(x), y=uv

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4.2 Chain Rule

The Chain Rule

  • Assume that there are functions f(x),\ g(x),\ h(x) where h(x)=f(g(x)). Hence,

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

i) If f(x)=c where c is a constant, then f^{\prime}(x)=0 (Constant function)

ii) If f(x)=k \cdot g(x), then f^{\prime}(x)=k \cdot g^{\prime}(x) (Multiple)

iii) If f(x)=g(x)+h(x), then f^{\prime}(x)=g^{\prime}(x)+h^{\prime}(x) (Sum)

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