# How do you use the chain rule to differentiate y=((5x^5-3)/(-3x^3+1))^3?

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Feb 24, 2018

$y = {\left(\frac{5 {x}^{2} - 3}{- 3 {x}^{3} + 1}\right)}^{3}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {\left(\frac{5 {x}^{2} - 3}{- 3 {x}^{3} + 1}\right)}^{2} \times \frac{d \left(\frac{5 {x}^{2} - 3}{- 3 {x}^{3} + 1}\right)}{\mathrm{dx}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {\left(\frac{5 {x}^{2} - 3}{- 3 {x}^{3} + 1}\right)}^{2} \times \left[\frac{\left(5 {x}^{2} - 3\right) ' \left(- 3 {x}^{3} + 1\right) - \left(5 {x}^{2} - 3\right) \left(- 3 {x}^{3} + 1\right) '}{- 3 {x}^{3} + 1} ^ 2\right]$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 3 {\left(\frac{5 {x}^{2} - 3}{- 3 {x}^{3} + 1}\right)}^{2} \times \left[\frac{10 x \left(- 3 {x}^{3} + 1\right) - \left(5 {x}^{2} - 3\right) \left(- 9 {x}^{2}\right)}{- 3 {x}^{3} + 1} ^ 2\right]$

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