Question

Use the Quotient Rule to find the UNSIMPLIFIED version of the derivative. R(w)=3w+w^4/ 2w^2+1.

Answer 1

From your previous questions, you know

(3w + w⁴)' = 3 + 4w³

(2w² + 1)' = 4w

So by the quotient rule,

R'(w) = [ (2w² + 1)•(3w + w⁴)' - (3w + w⁴)•(2w² + 1)' ] / (2w² + 1)²

That is, the quotient rule gives

R'(w) = [ (denominator)•(derivative of numerator) - (numerator)•(derivative of denominator) ] / (denominator)²

I'm not entirely sure what is meant by "unsimplified". Technically, you could stop here. But since you already know the component derivatives, might as well put them to use:

R'(w) = [ (2w² + 1)•(3 + 4w³) - (3w + w⁴)•(4w) ] / (2w² + 1)²