I have a math problem where I am required to find the derivative of a function with the limitations of not being allowed to use the Product or Quotient Rule of Differentiation.
The problem looks like this:
$$h(x) = \frac{4-x^6}{3x^{-2}}$$
I have tried a variety of routes but always end up with results that seem to require the use of the Product or Quotient Rule.
For example, my latest try looks like this:
$$h(x) = \frac{4-x^6}{3x^{-2}}$$
$$h(x) = \frac{4}{3x^{-2}} - \frac{x^6}{3x^{-2}}$$
$$h(x) = \frac{4x^2}{3} - \frac{x^8}{3}$$
(From this step, I figured I could just use the Difference Rule of Differentiation, like this:)
$$h'(x) = \frac{d}{dx}\left(\frac{4x^2}{3}\right) - \frac{d}{dx}\left(\frac{x^8}{3}\right)$$
But wouldn't this actually end up using the Product -or- Quotient Rule? Like this:
$$h'(x) = \frac{d}{dx}\left(\frac{4}{3}(x^2)\right) - \frac{d}{dx}\left(\frac{1}{3}(x^8)\right)$$
Is there another route I can take with this type of problem that would avoid using the Product or Quotient Rule of Differentiation?