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?
Best Answer
Most likely you are allowed to use the Constant Multiple Rule, i.e. $(cf(x))'=c(f(x))'$, where $c$ is a constant.