What is the derivative of (4-x)/x^(1/3)(6-x)^(2/3)?

Nov 22, 2016

Ew. If we assume that function is $f \left(x\right)$

f'(x) = (4x^2−16x−24)/(3*(6-x)^(1/3)x^(4/3)

Explanation:

Break up the equation and tackle it piece by piece.

I'll let you do the calculations but i'll simplify how it looks:

Let $a = \left(4 - x\right)$
Let $b = \left({x}^{\frac{1}{3}}\right)$
Let $c = {\left(6 - x\right)}^{\frac{2}{3}}$

$f ' \left(\frac{a}{b}\right) = \frac{\left[\frac{d}{\mathrm{dx}} \left(a\right) \cdot \left(b\right)\right] - \left[\frac{d}{\mathrm{dx}} \left(b\right) \cdot \left(a\right)\right]}{b} ^ 2$

Then use the product rule to finish it.

$f ' \left(x\right) = f ' \left(\frac{a}{b}\right) \cdot \left(c\right) + \left(\frac{a}{b}\right) \cdot \frac{d}{\mathrm{dx}} \left(c\right)$