Question

Дана функция y=(4x^2−4x+1)^3 Найдите y′′(x). В ответ введите значение y′′(1).

Answer 1

y = (2x - 1)⁶

y' = 12(2x-1)⁵

y'' = 120(2x-1)⁴

y''(1) = 120(2-1)⁴ = 120

Ответ: 120

Answer 2

To find the second derivative of the function $y = (4x^2 - 4x + 1)^3$, we'll need to take the derivative twice with respect to $x$.

Step 1: Find the first derivative, $y'$ (the derivative of $y$ with respect to $x$): $y' = \frac{d}{dx}[(4x^2 - 4x + 1)^3]$

To calculate this derivative, we can use the chain rule. If we have a function $f(u) = u^n$, then $f'(u) = n \cdot u^{n-1}$.

Applying the chain rule, we have: $y' = 3(4x^2 - 4x + 1)^2 \cdot \frac{d}{dx}(4x^2 - 4x + 1)$

Now, we need to find the derivative $\frac{d}{dx}(4x^2 - 4x + 1)$: $\frac{d}{dx}(4x^2 - 4x + 1) = 8x - 4$

Putting it back into the expression for $y'$: $y' = 3(4x^2 - 4x + 1)^2 \cdot (8x - 4)$

Step 2: Find the second derivative, $y''$ (the derivative of $y'$ with respect to $x$): $y'' = \frac{d}{dx}\left[3(4x^2 - 4x + 1)^2 \cdot (8x - 4)\right]$

Again, we need to use the chain rule. The derivative of $3(4x^2 - 4x + 1)^2$ is found as we did in Step 1, and the derivative of $(8x - 4)$ is simply 8.

So, we get: $y'' = 2 \cdot 3(4x^2 - 4x + 1) \cdot (8) = 48(4x^2 - 4x + 1)$

Now, to find $y''(1)$, we substitute $x = 1$ into the expression we just obtained: $y''(1) = 48(4(1)^2 - 4(1) + 1) = 48(4 - 4 + 1) = 48(1) = 48$

Therefore, the value of $y''(1)$ is $48$.