Question

Y' =(х÷(х-1)²)' найти производную

Answer 1

$y`= (\frac{x}{(x-1) ^{2} })`=[ (\frac{u}{v})`= \frac{u`\cdot v- u\cdot v`}{v ^{2} }]= \frac{x`(x-1) ^{2}-x\cdot((x-1) ^{2})` }{((x-1) ^{2})^{2} }= \\ =\frac{1\cdot (x-1) ^{2}-x\cdot2(x-1)\cdot (x-1)` }{(x-1) ^{4} } =\frac{(x-1)(x-1-2x) }{(x-1) ^{4} } = -\frac{(x+1) }{(x-1) ^{3} }$

Answer 2

Problem Statement

We are given the function y = (x/(x-1)² and we need to find its derivative.

Solution

To find the derivative of the given function, we can use the power rule and the quotient rule.

The power rule states that if we have a function of the form f(x) = x^n, then its derivative is given by f'(x) = n * x^(n-1).

The quotient rule states that if we have a function of the form f(x) = g(x)/h(x), then its derivative is given by f'(x) = (g'(x) * h(x) - g(x) * h'(x))/(h(x))^2.

Let's find the derivative of the given function step by step:

Step 1: Identify the numerator and denominator of the function. - Numerator: x - Denominator: (x-1)²

Step 2: Apply the power rule to the numerator. - The derivative of x is 1.

Step 3: Apply the quotient rule to the denominator. - The derivative of (x-1)² is 2(x-1) * (1) = 2(x-1).

Step 4: Apply the quotient rule to the entire function. - The derivative of the function is given by (1 * (x-1)² - x * 2(x-1))/(x-1)⁴.

Simplifying the expression, we get: - (x-1)² - 2x(x-1)/(x-1)⁴.

Further simplifying, we get: - (x-1) - 2x/(x-1)³.

Therefore, the derivative of the function y = (x/(x-1)² is (x-1) - 2x/(x-1)³.

Please let me know if I can help you with anything else.