# How do you differentiate y=((2x)/(x-1))((3x+4)/(5x^2-7))?

Nov 1, 2017

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x}{x - 1} \frac{3 x + 4}{5 {x}^{2} - 7} \left(\frac{1}{x} - \frac{1}{x - 1} + \frac{3}{3 x + 4} - \frac{10 x}{5 {x}^{2} - 7}\right)$

#### Explanation:

I would use logarithmic differentiation.

Taking the natural logarithm of both sides, we get:

$\ln y = \ln \left(\frac{2 x}{x - 1}\right) \left(\frac{3 x + 4}{5 {x}^{2} - 7}\right)$

$\ln y = \ln \left(2 x\right) - \ln \left(x - 1\right) + \ln \left(3 x + 4\right) - \ln \left(5 {x}^{2} - 7\right)$

Now differentiate.

$\frac{1}{y} \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = \frac{1}{x} - \frac{1}{x - 1} + \frac{3}{3 x + 4} - \frac{10 x}{5 {x}^{2} - 7}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = y \left(\frac{1}{x} - \frac{1}{x - 1} + \frac{3}{3 x + 4} - \frac{10 x}{5 {x}^{2} - 7}\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x}{x - 1} \frac{3 x + 4}{5 {x}^{2} - 7} \left(\frac{1}{x} - \frac{1}{x - 1} + \frac{3}{3 x + 4} - \frac{10 x}{5 {x}^{2} - 7}\right)$

Hopefully this helps!