Question

Y=e^x*(1+x^2) Решите производную функцию.

Answer 1

Y`=(e^x)`*(1+x^2)+(e^x)*(1+x^2)`
Y`=e^x*(1+x^2)+e^x*(2x)=e^x+x^2*e^x+2xe^x=e^x(x^2+2x+1)

Answer 2

To find the derivative of the given function $y = e^x(1+x^2)$ with respect to $x$, we will use the product rule and the chain rule.

The product rule states that if $y = u \cdot v$, then the derivative of $y$ with respect to $x$ is given by $\frac{{dy}}{{dx}} = u' \cdot v + u \cdot v'$, where $u'$ and $v'$ are the derivatives of $u$ and $v$ with respect to $x$ respectively.

Let's proceed with the differentiation:

$u = e^x$ (exponential function)
$u' = \frac{{d}}{{dx}}(e^x) = e^x$ (derivative of exponential function)

$v = 1 + x^2$
$v' = \frac{{d}}{{dx}}(1 + x^2) = 2x$ (derivative of the polynomial $x^2$)

Now apply the product rule:

$\frac{{dy}}{{dx}} = e^x \cdot (1 + x^2) + e^x \cdot 2x$

Simplify:

$\frac{{dy}}{{dx}} = e^x \cdot (1 + x^2 + 2x)$

So, the derivative of the function $y = e^x(1+x^2)$ with respect to $x$ is $\frac{{dy}}{{dx}} = e^x \cdot (1 + x^2 + 2x)$.