Question

determine the derivative of f(x) = |e^x + x|.

Explanation:

Step1: Recall chain - rule for derivative of absolute - value function

The derivative of $|u|$ with respect to $x$ is $\frac{u}{|u|}\cdot u'$ where $u = e^{x}+x$ and $u'$ is the derivative of $u$ with respect to $x$.

Step2: Find the derivative of $u = e^{x}+x$

The derivative of $e^{x}$ with respect to $x$ is $e^{x}$ and the derivative of $x$ with respect to $x$ is $1$. So, $u'=\frac{d}{dx}(e^{x}+x)=e^{x} + 1$.

Step3: Apply the chain - rule formula

Using the formula $\frac{d}{dx}|u|=\frac{u}{|u|}\cdot u'$, substituting $u = e^{x}+x$ and $u'=e^{x}+1$, we get $f'(x)=\frac{e^{x}+x}{|e^{x}+x|}\cdot(e^{x}+1)$.

Answer:

$f'(x)=\frac{e^{x}+x}{|e^{x}+x|}\cdot(e^{x}+1)$