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1. Prove that the derivative of tan(x) is sec²(x) directly from the definition of the derivative. You may not use the formula for the derivatives of sine or cosine. Hint: use the sum formula: and the identity: tan(x + y) = tan(x) + tan(y) 1 - tan(x) tan(y) 1 + tan² (x) = sec²(x).

1. Prove that the derivative of tan(x) is sec²(x) directly from the definition of the derivative. You may not use the formula for the derivatives of sine or cosine. Hint: use the sum formula: and the identity: tan(x + y) = tan(x) + tan(y) 1 - tan(x) tan(y) 1 + tan² (x) = sec²(x).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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USE THE SUM FORMULA
1. Prove that the derivative of tan(x) is sec²(x) directly from the definition of the derivative. You may not use the formula for the derivatives of sine or cosine. Hint: use the sum formula: and the identity: tan(x + y) = tan(x) + tan(y) 1 tan(x) tan(y) 1 + tan²(x) = sec²(x).
Transcribed Image Text:1. Prove that the derivative of tan(x) is sec²(x) directly from the definition of the derivative. You may not use the formula for the derivatives of sine or cosine. Hint: use the sum formula: and the identity: tan(x + y) = tan(x) + tan(y) 1 tan(x) tan(y) 1 + tan²(x) = sec²(x).
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