We now introduce the rule for the division. Try to derive a general formula from the examples you see in the following table:
| $$f (x)$$ | $$f'(x)$$ |
| $$\dfrac{x-1}{x}$$ | $$\dfrac{(1)x-(x-1)\cdot 1}{x^2}=\dfrac{1}{x^2}$$ |
| $$\dfrac{x^3}{x-2}$$ | $$\dfrac{3x^2(x-2)-x^3\cdot 1}{(x-2)^2}$$ |
| $$\dfrac{x}{x+2}$$ | $$\dfrac{1(x+2)-x \cdot 1}{(x+2)^2}$$ |
| $$\dfrac{3x^5}{2x+1}$$ | $$\dfrac{15x^5(2x+1)-3x^5(2)}{(2x+1)^2}$$ |
| $$\dfrac{g(x)}{h(x)}$$ | ? |
If you have been able to deduce the rule of the division, verify if it is the same as the one we present in what follows:
The derivative of the division of two functions is the derivative of the dividend times the divisor minus the dividend times the derivative of the divisor and divided by the square of the divisor. Mathematically it is undoubtedly clearer: $$$f(x)=\frac{g(x)}{h(x)} \Rightarrow f'(x)=\frac{g'(x)h(x)-g(x)h'(x)}{h^2(x)}$$$
Let's see some examples:
Let $$f(x)=\dfrac{x^2+x}{3x-1}$$
We identify $$g (x) =x^2+x$$ and $$h (x) = 3x-1$$. Let's apply the rule of the quotient,$$$f'(x)=\frac{(2x+1)(3x-1)-(x^2+x)3}{(3x-1)^2}$$$
Now a well-known example $$f(x)=\dfrac{x+2}{x^5}$$
Now $$g(x)=x+2$$ and $$h(x)=x^5$$. Let's apply the rule of the quotient,$$$f'(x)=\frac{1\cdot x^5-(x+2)5x^4}{(x^5)^2}$$$