Question

How do you differentiate `f(x)=((18x)/(4+x^2))` using the quotient rule?

Answer 1

The derivative is `(18 ( 4-x^2)) / ((4+x^2)^2)`

Explanation:

The quotient rule states that

`d/dx(f(x)/g(x)) = (f'(x)*g(x) - f(x)*g'(x))/g^2(x) color(white)(XXXXX) ( § )`

Let's calculate all the quantities we need:

• Since `f(x)=18x`, we have that `f'(x)=18`.
• Since `g(x) = 4+x^2`, we have that `g'(x)=2x`.
• As for `g(x)`, we'll just write `(4+x^2)^2`.

So, if we substitute all these functions in `(§)`, we get

`d/dx(f(x)/g(x)) =(18*(4+x^2) -18x* 2x ) / ((4+x^2)^2)`

we can easily expand the numerator into

`18*4 + 18x^2 - 36x^2 = 72 - 18x^2 = 18 ( 4-x^2)`

So, the expression becomes

`d/dx(f(x)/g(x)) = (18 ( 4-x^2)) / ((4+x^2)^2)`