Question

Y = 3x^2-1/x-1 , y" производные высших порядков

Answer 1

Y = 3x^2-1/x-1

У'=6х+ 1/х^2

У''=6 -2/х^3

Answer 2

To find the second derivative of the given function Y with respect to x, we need to apply differentiation twice. Let's start step by step:

Given function: Y = (3x^2 - 1) / (x - 1)

Step 1: Find the first derivative (y') of Y with respect to x. To find y', we use the quotient rule of differentiation:

(u/v)' = (u'v - uv') / v^2

where u = (3x^2 - 1) and v = (x - 1).

Now, applying the quotient rule:

Y' = [(2 * 3x * 1) - (3x^2 - 1) * 1] / (x - 1)^2 = (6x - 3x^2 + 1) / (x - 1)^2

Step 2: Find the second derivative (y") of Y with respect to x. To find y", we differentiate Y' (which we found in step 1) using the standard differentiation rules:

Y" = d/dx [(6x - 3x^2 + 1) / (x - 1)^2]

To simplify this expression, we need to apply the quotient rule again:

Let u = (6x - 3x^2 + 1) and v = (x - 1)^2

Then, u' = 6 - 6x (derivative of 6x), and v' = 2(x - 1) (derivative of (x - 1)^2).

Now, applying the quotient rule:

Y" = [(u'v - uv') / v^2]' = [(6 - 6x)(x - 1)^2 - (6x - 3x^2 + 1) * 2(x - 1)] / (x - 1)^4

Simplifying further:

Y" = [6(x - 1)^2 - 6x(x - 1)^2 - 2(6x - 3x^2 + 1)(x - 1)] / (x - 1)^4

Y" = [6(x - 1)^2 - 6x(x^2 - 2x + 1) - 2(6x^2 - 6x + 1)(x - 1)] / (x - 1)^4

Y" = [6(x^2 - 2x + 1) - 6x(x^2 - 2x + 1) - 2(6x^2 - 6x + 1)(x - 1)] / (x - 1)^4

Y" = [6x^2 - 12x + 6 - 6x^3 + 12x^2 - 6x - 2(6x^3 - 6x^2 + x - 1)] / (x - 1)^4

Y" = [6x^2 - 12x + 6 - 6x^3 + 12x^2 - 6x - 12x^3 + 12x^2 - 2x + 2] / (x - 1)^4

Y" = (-18x^3 + 30x^2 - 20x + 8) / (x - 1)^4

So, the second derivative of Y with respect to x is:

Y" = (-18x^3 + 30x^2 - 20x + 8) / (x - 1)^4