Find the derivative of f(x) = e^(2x) at x = 0.

₹0.0

Question: Find the derivative of f(x) = e^(2x) at x = 0.

Options:

Correct Answer: 2

Solution:

f\'(x) = 2e^(2x). At x = 0, f\'(0) = 2e^0 = 2.

Find the derivative of f(x) = e^(2x) at x = 0.

Find the derivative of f(x) = e^(2x) at x = 0.

Correct Answer: 2

Step 1: Identify the function we need to differentiate, which is f(x) = e^(2x).
Step 2: Use the chain rule to find the derivative. The derivative of e^(u) is e^(u) * du/dx, where u = 2x.
Step 3: Calculate du/dx. Since u = 2x, the derivative du/dx = 2.
Step 4: Apply the chain rule: f'(x) = e^(2x) * 2 = 2e^(2x).
Step 5: Now, we need to find the derivative at x = 0. Substitute x = 0 into f'(x).
Step 6: Calculate f'(0) = 2e^(2*0) = 2e^0.
Step 7: Since e^0 = 1, we have f'(0) = 2 * 1 = 2.

Differentiation of Exponential Functions – Understanding how to differentiate functions of the form f(x) = e^(kx) where k is a constant.
Evaluation of Derivatives at Specific Points – Calculating the value of the derivative at a given point, in this case, x = 0.