DerivativesOfTrigs

# DerivativesOfTrigs - DERIVATIVES OF TRIGONOMETRIC FUNCTIONS...

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The Sine Function We begin with the derivative of the sine function using the original definition of the derivative: f ' x = lim h 0 f x h  − f x h In applying this definition we begin with the traditional “sinco” expansion in the top left term and then proceed to do some algebraic manipulation leading to the special limit for the sine function, lim h 0 sin h h = 1. D x sin x = lim h 0 sin x h  − sin x h = lim h 0 sin x cos h cos x sin h  − sin x h = lim h 0 cos x  sin h h sin x  cos h 1 h = lim h 0 cos x sin h h  − sin x 1 cos h h As h 0 the first term in the limit presents no difficulty as we can apply the special limit. The second term becomes − sin x 0 0 which is undefined. We apply the old conjugate trick and then we can proceed easily with the limit evaluation. D x sin x = lim h 0 cos x sin h h  − sin x 1 cos h h 1 cos h 1 cos h In the second term we replace 1 cos 2 h with sin 2 h , and then put one sin h over the h to avoid the poisonous zero over zero conundrum. D x sin x = lim h 0 cos x sin h h  − sin x sin h h sin h 1 cos h We are now ready to evaluate the limit. D

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DerivativesOfTrigs - DERIVATIVES OF TRIGONOMETRIC FUNCTIONS...

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