Question

Find The Derivatives Of All Orders Of The Following Function. Y=8 / x⁴ −4 / 9x2−9x

Y′= ______
Y′′=________
Y′′′= __________
Y(4)=__________
Y(5)=___________
All Derivatives Of Order Greater than 5 are ________

Answer 1

To find the derivatives of the function y = 8 / x⁴ - 4 / 9x² - 9x, use the power rule and quotient rule. The first derivative is -32 / x⁵ + (72x - 4)/(9x³). The second derivative is 160 / x⁶ - (216x² + 72) / (9x⁴). The third derivative is -960 / x⁷ + (864x⁴ - 432x) / (9x⁵). Substitute specific values of x to find Y(4) and Y(5).

Step-by-step explanation:

To find the derivatives of all orders of the function y = 8 / x⁴ - 4 / 9x² - 9x, we can use the power rule and the quotient rule.

First, let's find the first derivative, y'.

1. Using the power rule, differentiate the first term: d/dx (8 / x⁴) = -32 / x⁵.
2. Using the quotient rule, differentiate the second term: d/dx (-4 / 9x² - 9x) = (72x - 4)/(9x³).
3. Combining the derivatives of both terms, we get: y' = -32 / x⁵ + (72x - 4)/(9x³).

Next, let's find the second derivative, y''.

1. Differentiate y' using the power rule and quotient rule: y'' = d/dx (-32 / x⁵ + (72x - 4)/(9x³)) = 160 / x⁶ - (216x² + 72) / (9x⁴).

The third derivative, y''', can be found by differentiating y'' using the power rule and quotient rule: y''' = d/dx (160 / x⁶ - (216x² + 72) / (9x⁴)) = -960 / x⁷ + (864x⁴ - 432x) / (9x⁵).

To find Y(4) and Y(5), substitute x = 4 and x = 5 respectively in the original function y = 8 / x⁴ - 4 / 9x² - 9x. For any derivatives of order greater than 5, substitute the value of x in the corresponding derivative function.