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Misc 17 - Find derivative: sin x + cos x / sin x - cos x - Derivatives by formula - sin & cos

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  1. Chapter 13 Class 11 Limits and Derivatives
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Misc 17 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sin﷮x + cos﷮x﷯﷯﷮ sin﷮x − cos﷮x﷯﷯﷯ Let f (x) = sin﷮x + cos﷮x﷯﷯﷮ sin﷮x − cos﷮x﷯﷯﷯ Let u = sin x + cos x & v = sin x – cos x ∴ f(x) = 𝑢﷮𝑣﷯ So, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ Using quotient rule f’(x) = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ Finding u’ & v’ u = sin x + cos x u’ = (sin x + cos x)’ = (sin x)’ + (cos x)’ = cos x – sin x v = sin x – cos x v’= (sin x – cos x)’ = (sin x)’ – (cos x)’ = cos x – ( – sin x) = cos x + sin x Now, f’(x) = 𝑢﷮𝑣﷯﷯﷮′﷯ = 𝑢﷮′﷯𝑣 − 𝑣﷮′﷯𝑢﷮ 𝑣﷮2﷯﷯ = ( cos﷮𝑥 − sin﷮𝑥) ( sin﷮𝑥 − cos﷮𝑥) − ( cos﷮𝑥 + sin﷮𝑥) ( sin﷮𝑥 + cos﷮𝑥)﷯﷯﷯﷯﷯﷯﷯﷯﷮ ( sin﷮x −co𝑠 𝑥﷯)﷮2﷯﷯ = − ( sin﷮𝑥 − cos﷮𝑥) ( sin﷮𝑥 − cos﷮𝑥) − ( sin﷮𝑥 + cos﷮𝑥) ( sin﷮𝑥 + cos﷮𝑥)﷯﷯﷯﷯﷯﷯﷯﷯﷮ ( sin﷮x − co𝑠 𝑥﷯)﷮2﷯﷯ = −( sin﷮x − co𝑠 𝑥﷯)﷮2﷯ − ( sin﷮x + co𝑠 𝑥﷯)﷮2﷯﷮ ( sin﷮x − co𝑠 𝑥﷯)﷮2﷯﷯ = − [( sin2﷮𝑥 + cos2﷮𝑥 − 2 sin﷮𝑥 cos﷮𝑥) + (𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥 + 2𝑠𝑖𝑛𝑥 cos﷮𝑥)]﷯﷯﷯﷯﷯﷮ ( sin﷮x − co𝑠 𝑥﷯)﷮2﷯﷯ = − ( 2𝑠𝑖𝑛2𝑥 + 2𝑐𝑜𝑠2𝑥 − 0)﷮ ( sin﷮x − co𝑠 𝑥﷯)﷮2﷯﷯ = −2 (𝒔𝒊𝒏𝟐𝒙 + 𝒄𝒐𝒔𝟐𝒙)﷮ ( sin﷮x − co𝑠 𝑥﷯)﷮2﷯﷯ = −2 (𝟏)﷮ ( sin﷮x − co𝑠 𝑥﷯)﷮2﷯﷯ = −𝟐 ﷮ ( 𝒔𝒊𝒏﷮𝐱 − 𝒄𝒐𝒔 𝒙﷯)﷮𝟐﷯﷯

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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