Question

hw_2.1_basic derivative rules due sep 14, 2025 11:59 pm. 9. submit answer practice similar attempt 1: 10 attempts remaining. if $f(x)=4e^{x}+2$, find $f(x)$. $f(x)=$

Explanation:

Step1: Recall derivative rules

The derivative of a sum of functions is the sum of the derivatives, i.e., if $y = u + v$, then $y'=u'+v'$. Also, the derivative of $e^{x}$ is $e^{x}$ and the derivative of a constant $C$ is 0.

Step2: Differentiate each term

For the function $f(x)=4e^{x}+2$, let $u = 4e^{x}$ and $v = 2$. The derivative of $u$ with respect to $x$: Using the constant - multiple rule $(cf(x))'=cf'(x)$ where $c = 4$ and $f(x)=e^{x}$, we have $u'=4\frac{d}{dx}(e^{x})=4e^{x}$. The derivative of $v$ with respect to $x$ is $v'=\frac{d}{dx}(2)=0$.

Step3: Find $f'(x)$

By the sum - rule of derivatives $f'(x)=u'+v'$. Substituting $u' = 4e^{x}$ and $v' = 0$ into the formula, we get $f'(x)=4e^{x}+0 = 4e^{x}$.

Answer:

$4e^{x}$