# How do you find the derivative of (x^8(x-23)^(1/2))/(27x^6(4x-6)^8)?

Dec 8, 2017

=(x^8(432x^5(4x-6)^8(x-23)^(1/2)+(x-23)^(-1/2)-2x^4(x-23)^(1/2)(54x^5(3(4x-6)^8+16x(4x-6)^7))))/(2(27x^6(4x-8)^8)^2

#### Explanation:

$y = \frac{{x}^{8} {\left(x - 23\right)}^{\frac{1}{2}}}{27 {x}^{6} {\left(4 x - 6\right)}^{8}} = f \frac{x}{g} \left(x\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{f ' \left(x\right) g \left(x\right) - f \left(x\right) g ' \left(x\right)}{f {\left(x\right)}^{2}}$

$f \left(x\right) = {x}^{8} {\left(x - 23\right)}^{\frac{1}{2}} = h \left(x\right) j \left(x\right)$
$f ' \left(x\right) = h ' \left(x\right) j \left(x\right) + h \left(x\right) j ' \left(x\right)$
$h \left(x\right) = {x}^{8}$
$h ' \left(x\right) = 8 {x}^{7}$
$j \left(x\right) = {\left(x - 23\right)}^{\frac{1}{2}}$
$j ' \left(x\right) = {\left(x - 23\right)}^{- \frac{1}{2}} / 2$
$f ' \left(x\right) = 8 {x}^{7} {\left(x - 23\right)}^{\frac{1}{2}} + \frac{{x}^{8} {\left(x - 23\right)}^{- \frac{1}{2}}}{2} = \frac{16 {x}^{7} {\left(x - 23\right)}^{\frac{1}{2}} + {x}^{8} {\left(x - 23\right)}^{- \frac{1}{2}}}{2} = \frac{16 {x}^{7} {\left(x - 23\right)}^{\frac{1}{2}} + {x}^{8} {\left(x - 23\right)}^{- \frac{1}{2}}}{2}$

$g \left(x\right) = 27 {x}^{6} {\left(4 x - 6\right)}^{8} = a \left(x\right) s \left(x\right)$
$g ' \left(x\right) = a ' \left(x\right) s \left(x\right) + a \left(x\right) s ' \left(x\right)$
$a \left(x\right) = 27 {x}^{6}$
$a ' \left(x\right) = 162 {x}^{5}$
$s \left(x\right) = {\left(4 x - 6\right)}^{8}$
$s ' \left(x\right) = 4 \cdot 8 \cdot {\left(4 x - 6\right)}^{7} = 32 {\left(4 x - 6\right)}^{7}$
$g ' \left(x\right) = 162 {x}^{5} {\left(4 x - 6\right)}^{8} + 27 {x}^{6} 32 {\left(4 x - 6\right)}^{7} = 162 {x}^{5} {\left(4 x - 6\right)}^{8} + 864 {x}^{6} {\left(4 x - 6\right)}^{7} = 54 {x}^{5} \left(3 {\left(4 x - 6\right)}^{8} + 16 x {\left(4 x - 6\right)}^{7}\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{27 {x}^{6} {\left(4 x - 6\right)}^{8} \left(\frac{16 {x}^{7} {\left(x - 23\right)}^{\frac{1}{2}} + {x}^{8} {\left(x - 23\right)}^{- \frac{1}{2}}}{2}\right) - {x}^{8} {\left(x - 23\right)}^{\frac{1}{2}} \left(54 {x}^{5} \left(3 {\left(4 x - 6\right)}^{8} + 16 x {\left(4 x - 6\right)}^{7}\right)\right)}{27 {x}^{6} {\left(4 x - 6\right)}^{8}} ^ 2$
$= \frac{\left(\frac{27 {x}^{6} {\left(4 x - 6\right)}^{8} 16 {x}^{7} {\left(x - 23\right)}^{\frac{1}{2}} + {x}^{8} {\left(x - 23\right)}^{- \frac{1}{2}}}{2}\right) - {x}^{8} {\left(x - 23\right)}^{\frac{1}{2}} \left(54 {x}^{5} \left(3 {\left(4 x - 6\right)}^{8} + 16 x {\left(4 x - 6\right)}^{7}\right)\right)}{27 {x}^{6} {\left(4 x - 6\right)}^{8}} ^ 2$
$= \frac{27 {x}^{6} {\left(4 x - 6\right)}^{8} 16 {x}^{7} {\left(x - 23\right)}^{\frac{1}{2}} + {x}^{8} {\left(x - 23\right)}^{- \frac{1}{2}} - 2 {x}^{8} {\left(x - 23\right)}^{\frac{1}{2}} \left(54 {x}^{5} \left(3 {\left(4 x - 6\right)}^{8} + 16 x {\left(4 x - 6\right)}^{7}\right)\right)}{2 {\left(27 {x}^{6} {\left(4 x - 6\right)}^{8}\right)}^{2}}$
=(x^8(432x^5(4x-6)^8(x-23)^(1/2)+(x-23)^(-1/2)-2x^4(x-23)^(1/2)(54x^5(3(4x-6)^8+16x(4x-6)^7))))/(2(27x^6(4x-8)^8)^2