Derivative of y=e^(3x) - Explained with Chain Rule

Differentiating y=e^(3x) using the Chain Rule

To find the derivative of y=e^(3x), we can employ the chain rule of differentiation.

1. Let u = 3x. This allows us to simplify the expression.
2. Find du/dx: The derivative of u with respect to x is 3 (du/dx = 3).
3. Apply the Chain Rule: The chain rule states that dy/dx = (dy/du) * (du/dx).
4. Differentiate e^u: We know that the derivative of e^u with respect to u is simply e^u (dy/du = e^u).
5. Substitute u back in: Replace u with 3x in our derivative: dy/du = e^(3x).
6. Combine the components: Now, using the chain rule: dy/dx = (dy/du) * (du/dx) = e^(3x) * 3.

Therefore, the derivative of y=e^(3x) is:

dy/dx = 3e^(3x)