question\ngiven $f(x)$ below, find $f(x)$.\n$f(x)=int_{ln(x)}^{5x}e^{t}dt$\nprovide your answer…

Explanation:

Step1: Apply the fundamental theorem of calculus and chain - rule

If $F(t)$ is an antiderivative of $e^{t}$, i.e., $F^\prime(t)=e^{t}$, then $f(x)=F(5x)-F(\ln(x))$.

Step2: Differentiate using the chain - rule

By the chain - rule, if $y = F(u)$ and $u$ is a function of $x$, then $\frac{dy}{dx}=F^\prime(u)\cdot u^\prime$. For $y = F(5x)$, $\frac{d}{dx}F(5x)=F^\prime(5x)\cdot\frac{d}{dx}(5x)=e^{5x}\cdot5$. For $y = F(\ln(x))$, $\frac{d}{dx}F(\ln(x))=F^\prime(\ln(x))\cdot\frac{d}{dx}(\ln(x))=e^{\ln(x)}\cdot\frac{1}{x}$. Since $e^{\ln(x)} = x$, we have $\frac{d}{dx}F(\ln(x))=x\cdot\frac{1}{x}=1$.

Step3: Calculate $f^\prime(x)$

$f^\prime(x)=\frac{d}{dx}(F(5x)-F(\ln(x)))=5e^{5x}-1$.

Answer:

$5e^{5x}-1$