differentiate x^2ex^2 W.R TO X

if y = uv, y' = u'v + uv'

if v = e^w, v' = e^w w'
here we have u=x^2, v=e^w, w=x^2

y' = 2x e^x^2 + x^2 (2x) e^x^2
= 2x e^x^2 (x^2 + 1)

To differentiate the function \(f(x) = x^2e^{x^2}\) with respect to \(x\), we use the product rule.

The product rule states that for two functions \(u(x)\) and \(v(x)\), the derivative of their product \(u(x)v(x)\) with respect to \(x\) is given by:

\[\frac{d}{dx} (u(x)v(x)) = u'(x)v(x) + u(x)v'(x)\]

Let's apply the product rule to \(f(x) = x^2e^{x^2}\):

\(u(x) = x^2\) and \(v(x) = e^{x^2}\)

Taking the derivatives of \(u(x)\) and \(v(x)\) gives:

\(u'(x) = 2x\)
\(v'(x) = 2xe^{x^2}\)

Now, we can plug these values into the product rule formula:

\[\frac{d}{dx} (x^2e^{x^2}) = (2x)(e^{x^2}) + (x^2)(2xe^{x^2})\]

Simplifying further, we get:

\[\frac{d}{dx} (x^2e^{x^2}) = 2xe^{x^2} + 2x^3e^{x^2}\]

Thus, the derivative of \(f(x) = x^2e^{x^2}\) with respect to \(x\) is \(2xe^{x^2} + 2x^3e^{x^2}\).