Question

G(x)=3/x find g'(x) a) by using the quotient rule: b) by first rewriting the expression using exponents and then differentiating.

Answer 1

a) The quotient rule is:

if: f(x) = 1/g(x)

Then:

`f'(x) = -(1)/((g(x))^2)*g'(x)`

In this case, we have:

G(x) = 1/x

Then we can write this as:

G(x) = 1/h(x) with h(x) = x, and h'(x) = 1

Using the above rule we get:

G'(x) = -(1/h(x)^2)*h'(x) = -1/x^2

b) For a function like:

f(x) = x^n

we have that:

`f'(x) = n*x^(n - 1)`

Here we can write G(x) = x^-1

Then we have n = -1

If we use the above rule, we get:

`G'(x) = (-1)*x^(-1-1) = -1*x^(-2) = (-1)/(x^2)`

So we got the same result using both methods.