differentiate f(x) = 3e^7x f(x)=

Explanation:

Step1: Recall chain - rule

The chain - rule states that if $y = f(g(x))$, then $y^\prime=f^\prime(g(x))\cdot g^\prime(x)$. For $y = 3e^{7x}$, let $u = 7x$, so $y = 3e^{u}$.

Step2: Differentiate outer function

The derivative of $y = 3e^{u}$ with respect to $u$ is $\frac{dy}{du}=3e^{u}$.

Step3: Differentiate inner function

The derivative of $u = 7x$ with respect to $x$ is $\frac{du}{dx}=7$.

Step4: Apply chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=3e^{u}$ and $\frac{du}{dx}=7$ and $u = 7x$ back in, we get $f^\prime(x)=3e^{7x}\cdot7$.

Answer:

$21e^{7x}$