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Find the derivative of (1+x)(1+ex2)

by RoRi

Find the derivative of the following function:

    \[ f(x) = (1+x)\left( 1+ e^{x^2} \right). \]

We can compute this derivative by multiplying out the expression for f(x) and using the product and chain rules. First, we multiply out the expression,

    \begin{align*} f(x) &= (1+x) \left(1+e^{x^2} \right) \\ & = 1 + x + e^{x^2} + xe^{x^2}. \end{align*}

Then, we take the derivative,

    \begin{align*} f'(x) &= 1 + 2x e^{x^2} + e^{x^2} + 2x^2 e^{x^2} \\ &= 1 + (1+2x + 2x^2)e^{x^2}. \end{align*}

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