Find the derivative of f(x) = tan(x) at x = π/4.

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Question: Find the derivative of f(x) = tan(x) at x = π/4.

Options:

Correct Answer: 1

Solution:

f\'(x) = sec^2(x). At x = π/4, f\'(π/4) = sec^2(π/4) = 2.

Find the derivative of f(x) = tan(x) at x = π/4.

Find the derivative of f(x) = tan(x) at x = π/4.

Correct Answer: 2

Step 1: Identify the function we need to differentiate. Here, f(x) = tan(x).
Step 2: Recall the derivative of tan(x). The derivative is f'(x) = sec^2(x).
Step 3: Substitute x = π/4 into the derivative. We need to find f'(π/4) = sec^2(π/4).
Step 4: Calculate sec(π/4). Since sec(x) = 1/cos(x), we find cos(π/4) = √2/2, so sec(π/4) = 1/(√2/2) = 2/√2 = √2.
Step 5: Now, square sec(π/4) to find sec^2(π/4). So, sec^2(π/4) = (√2)^2 = 2.
Step 6: Therefore, the derivative of f(x) = tan(x) at x = π/4 is f'(π/4) = 2.

Derivative of Trigonometric Functions – Understanding how to differentiate trigonometric functions, specifically the derivative of tan(x) which is sec^2(x).
Evaluating Functions at Specific Points – Calculating the derivative at a specific value, in this case, x = π/4.