Question

e3x find the derivatives

Answer 1

3e^{3x}

Answer 2

To find the derivative of $e^{3x}$, we use the chain rule.

Let $y = e^{3x}$. Let $u = 3x$. Then $y = e^u$.

According to the chain rule, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.

First, find $\frac{dy}{du}$: The derivative of $e^u$ with respect to $u$ is $e^u$. So, $\frac{dy}{du} = e^u$.

Next, find $\frac{du}{dx}$: The derivative of $3x$ with respect to $x$ is $3$. So, $\frac{du}{dx} = 3$.

Now, substitute these back into the chain rule formula: $\frac{dy}{dx} = e^u \cdot 3$

Finally, substitute $u = 3x$ back into the expression: $\frac{dy}{dx} = 3e^{3x}$

The derivative of $e^{3x}$ is $3e^{3x}$.

Explanation: To find the derivative of $e^{3x}$, we apply the chain rule. The general rule for differentiating $e^{ax}$ is $a e^{ax}$. Here, $a=3$, so the derivative is $3e^{3x}$.