Find the derivative of f(w)=w4/5+8w.

Question

Find the derivative of f(w)=w4/5+8w.

Enclose numerators and denominators in parentheses. For example (a−b)/(1+n).

Answer 1

f(w) = "w4/5" + 8w

could mean f(w) = (1/5)w^4 + 8w

then f'(w) = (4/5)w^3 + 8 = 4w^3/5 + 8 = (4w^3+40)/(5)

but the way it's written it might mean: f(w) = 4w/5 +8w = 44w/5

then f'(w) = 44/5 = (44)/(5) if you're into unnecessary parentheses

my guess is you want the first interpretation and answer (4w^3+40)/(5)

but there's a 3rd possibility

maybe you meant f(x) = x^4/(5+8x)

this would be right if you had just left out the parentheses, or the original problem left them out, which is very possible

then f(x) = x^4(5+8x)^-1 and

f'(x) = first term times derivative of the 2nd term + 2nd term times derivative of the 1st term

f'(x) = -8x^4/(5+8x)^2 + (1/(5+8x)(4x^3) = [-8x^4 +(4x^3)(5+8x)]/(5+8x)^2 = [-8x^4 +32x^4 + 20x^3]/(5+8x)^2

= (24x^4 +20x^3)/(64x^2 +80x +25)

that's complicated enough to get students make a mistake somewhere, so the instructor can take points off. It also requires a little more calculus than the 1st two interpretations of the problem, so maybe this 3rd interpretation is what you want. Plus the 3rd answer is more of the form you want in terms of parentheses.