Question

Производная y= (x+x^2)^x

Answer 1

Ответ: пользцемся правилом производной функция^функция

Пошаговое объяснение:

См. Фото

Answer 2

To find the derivative of the function $y = (x + x^2)^x$, we'll use the chain rule. The chain rule states that if we have a composition of functions, $y = u^v$, then the derivative with respect to $x$ is given by:

$\frac{{dy}}{{dx}} = v \cdot u^{v-1} \cdot \frac{{du}}{{dx}} + u^v \cdot \ln(u) \cdot \frac{{dv}}{{dx}}$

where $u$ and $v$ are both functions of $x$.

Let's differentiate $y = (x + x^2)^x$ step by step:

Step 1: Identify $u$ and $v$ In this case, $u = x + x^2$ and $v = x$.

Step 2: Find $\frac{{du}}{{dx}}$ and $\frac{{dv}}{{dx}}$ $\frac{{du}}{{dx}} = 1 + 2x$ $\frac{{dv}}{{dx}} = 1$

Step 3: Substitute into the chain rule formula $\frac{{dy}}{{dx}} = x \cdot (x + x^2)^{x-1} \cdot (1 + 2x) + (x + x^2)^x \cdot \ln(x + x^2) \cdot 1$

Simplified, the derivative is: $\frac{{dy}}{{dx}} = x \cdot (x + x^2)^{x-1} \cdot (1 + 2x) + (x + x^2)^x \cdot \ln(x + x^2)$

And that's the derivative of the given function.