Question

if $f(x)=14e^{x}+x^{e}$, find $f(x)$.

Explanation:

Step1: Recall derivative rules

The derivative of $e^x$ is $e^x$ and the power - rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$.

Step2: Differentiate $14e^x$

Using the constant - multiple rule $\frac{d}{dx}(cf(x))=c\frac{d}{dx}(f(x))$ where $c = 14$ and $f(x)=e^x$, we have $\frac{d}{dx}(14e^x)=14\frac{d}{dx}(e^x)=14e^x$.

Step3: Differentiate $x^e$

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$ with $n = e$, we get $\frac{d}{dx}(x^e)=ex^{e - 1}$.

Step4: Find $f'(x)$

By the sum - rule of differentiation $\frac{d}{dx}(u(x)+v(x))=\frac{d}{dx}(u(x))+\frac{d}{dx}(v(x))$, where $u(x)=14e^x$ and $v(x)=x^e$, we have $f'(x)=14e^x+ex^{e - 1}$.

Answer:

$f'(x)=14e^x+ex^{e - 1}$