First use the quotient rule.

f ' (x) = [((x)^{2} + 6)[(5(x)^{2}-7)((x)^{2 }- 2)]' - [(5(x)^{2}-7)((x)^{2} - 2)]((x)^{2 }+ 6)' ]/((x)^{2 }+ 6)^{2}

Now we need to know derivatives of g(x) = (5(x)^{2 }- 7)((x)^{2} - 2) and h(x) = (x)^{2} + 6.

Clearly h ' (x) = 2(x) and for the derivative of g(x) we use the product rule. g ' (x) = (10x)((x)^{2}-2) + (5(x)^{2} - 7)(2x)^{ }

So putting this together we get f ' (x) = [((x)^{2 }+ 6)[(10(x))((x)^{2 }- 2) + (5(x)^{2 }- 7)(2x)] - [(5(x)^{2}-7)((x)^{2} - 2)](2x)]/((x)^{2} + 6)^{2}

Plugging in 5 for x in the last equation gives 46.9199

So to sum up, we applied a quotient rule first to the whole thing and within that we used the product rule for the derivative of the numerator.

Hope this helps.

Philip