find f(x). f(x)=e^16x f(x)=

Explanation:

Step1: Identify the function type

The function $f(x)=e^{16x}$ is an exponential - composite function.

Step2: Apply the chain - rule

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. For $y = e^{u}$ and $u = 16x$, the derivative of $y = e^{u}$ with respect to $u$ is $\frac{dy}{du}=e^{u}$, and the derivative of $u = 16x$ with respect to $x$ is $\frac{du}{dx}=16$.

Step3: Calculate the derivative

By the chain - rule, $f^\prime(x)=\frac{dy}{du}\cdot\frac{du}{dx}=e^{u}\cdot16$. Substituting $u = 16x$ back in, we get $f^\prime(x)=16e^{16x}$.

Answer:

$16e^{16x}$