Question

differentiate. f(x)=5e^{6x} f(x)=□

Explanation:

Step1: Recall 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 a function of the form $y = ae^{bx}$ (where $a$ and $b$ are constants), we can consider $u = bx$ and $y = ae^{u}$.

Step2: Differentiate the outer function

The derivative of $y = ae^{u}$ with respect to $u$ is $\frac{dy}{du}=ae^{u}$. Here, $a = 5$ and $u = 6x$, so $\frac{dy}{du}=5e^{u}=5e^{6x}$.

Step3: Differentiate the inner function

The derivative of $u = 6x$ with respect to $x$ is $\frac{du}{dx}=6$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substituting $\frac{dy}{du}=5e^{6x}$ and $\frac{du}{dx}=6$ into the chain - rule formula, we get $\frac{dy}{dx}=5e^{6x}\cdot6$.

Answer:

$30e^{6x}$