Differentiate? F(y) = (1/y^2 - 5/y^4) (y +7y^3)

May 12, 2017

$\frac{\mathrm{dF} \left(y\right)}{\mathrm{dy}} = \left(- 2 {y}^{-} 3 + 20 {y}^{-} 5\right) \left(y + 7 {y}^{3}\right) + \left(\frac{1}{y} ^ 2 - \frac{5}{y} ^ 4\right) \left(1 + 21 {y}^{2}\right)$

Explanation:

Use the product rule:

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

Do the first derivative:

$\frac{d \left(\frac{1}{y} ^ 2 - \frac{5}{y} ^ 4\right)}{\mathrm{dy}} = \frac{d \left({y}^{-} 2 - 5 {y}^{-} 4\right)}{\mathrm{dy}} = - 2 {y}^{-} 3 + 20 {y}^{-} 5$

Substitute back into the product rule:

$\frac{\mathrm{dF} \left(y\right)}{\mathrm{dy}} = \left(- 2 {y}^{-} 3 + 20 {y}^{-} 5\right) \left(y + 7 {y}^{3}\right) + \left(\frac{1}{y} ^ 2 - \frac{5}{y} ^ 4\right) \frac{d \left(y + 7 {y}^{3}\right)}{\mathrm{dy}}$

Do the second derivative:

$\frac{d \left(y + 7 {y}^{3}\right)}{\mathrm{dy}} = 1 + 21 {y}^{2}$

Substitute back into the product rule:

$\frac{\mathrm{dF} \left(y\right)}{\mathrm{dy}} = \left(- 2 {y}^{-} 3 + 20 {y}^{-} 5\right) \left(y + 7 {y}^{3}\right) + \left(\frac{1}{y} ^ 2 - \frac{5}{y} ^ 4\right) \left(1 + 21 {y}^{2}\right)$