Question

How do you calculate the derivate for `f(x)=[(x+2)/(x+1)](2x-7)`?

Answer 1

Using a combination of Product Rule and Quotient Rule.

Explanation:

Product Rule:
If `f(x)=g(x)\cdot h(x)`,
then `f^\prime(x)=g(x)\cdot h\prime (x)+h(x)\cdot g\prime (x)`

Quotient Rule:
If `f(x)=\frac{g(x)}{h(x)}`
then `f^\prime(x)=\frac{h(x)\cdot g\prime (x)-g(x)\cdot h\prime (x)}{h(x)^2}`

Now, you can express `f(x)` as

`f(x)=\underbrace{g(x)\cdot h(x)}_\text{product rule}`,

where `g(x)=\underbrace{\frac{x+2}{x+1}}_\text{quotient rule}` and `h(x)=2x-7`

OR as

`f(x)=\underbrace{\frac{g(x)}{h(x)}}_\text{quotient rule}`,

where `g(x)=\underbrace{(x+2)(2x-7)}_\text{product rule}` and `h(x)=x+1`.

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