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).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Problem Statement:**
1. Prove that the derivative of \( \tan(x) \) is \( \sec^2(x) \) 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:*
\[
\tan(x+y) = \frac{\tan(x) + \tan(y)}{1 - \tan(x) \tan(y)}
\]
*and the identity:*
\[
1 + \tan^2(x) = \sec^2(x).
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4979c4fb-69aa-4561-8ea9-69969c4f32d0%2Fc15ac434-b475-4dff-821b-f20130994b6b%2Fq2fasmj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
1. Prove that the derivative of \( \tan(x) \) is \( \sec^2(x) \) 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:*
\[
\tan(x+y) = \frac{\tan(x) + \tan(y)}{1 - \tan(x) \tan(y)}
\]
*and the identity:*
\[
1 + \tan^2(x) = \sec^2(x).
\]
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