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Find the derivative of f(x) = tan(x) at x = π/4.
Find the derivative of f(x) = tan(x) at x = π/4.
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Practice Questions
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Q1
Find the derivative of f(x) = tan(x) at x = π/4.
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f'(x) = sec^2(x). At x = π/4, f'(π/4) = sec^2(π/4) = 2.
Questions & Step-by-step Solutions
1 item
Q
Q: Find the derivative of f(x) = tan(x) at x = π/4.
Solution:
f'(x) = sec^2(x). At x = π/4, f'(π/4) = sec^2(π/4) = 2.
Steps: 6
Show Steps
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.
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