1 x cot 2 x h 2 2 2 2 2 ᆴ ᆴ ᆴ ᆴ ᆴ ᆴ ᆴ

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Mathematics for Machine Technology
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Chapter 1 / Exercise 1
Mathematics for Machine Technology
Peterson/Smith
Expert Verified
1 x cot 2 x ' h 2 2 2 2 2 x sin x sin x cos x sin 2 x ' h 2 2 2 2 x 2 cos 2 x ' h
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Mathematics for Machine Technology
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Chapter 1 / Exercise 1
Mathematics for Machine Technology
Peterson/Smith
Expert Verified
1 x csc x F . 9 3 1 x csc 2 x 3 1 x cot 1 x csc x ' F 3 2 3 3 1 x csc 2 1 x csc 1 x cot 1 x csc x 3 x ' F 3 3 3 3 2 1 x csc 1 x cot x 2 3 x ' F 3 3 2
Find the derivative and simplify the result. EXERCISES : 3 x 4 5 sin ln x h . 1 3 2 x ln cos x f . 2 x 4 cos 2 x 4 sin x g . 3 x 2 cos x 4 sin 2 x 2 sin x cos 2 x F . 4 x cos 3 1 sin y . 5 3 x tan x sin x F . 6 y x tan y . 7 2 2 x 1 x 2 cot x F . 8 0 xy xy cot . 9 0 y csc x sec . 10 2 2
DIFFERENTIATION OF INVERSE TRIGONOMETRIC FUNCTIONS
TRANSCENDENTAL FUNCTIONS Kinds of transcendental functions: 1. logarithmic and exponential functions 2. trigonometric and inverse trigonometric functions 3. hyperbolic and inverse hyperbolic functions Note: Each pair of functions above is an inverse to each other.
The INVERSE TRIGONOMETRIC FUNCTIONS x. is sine whose angle the is y mean also This x sin y or x arcsin y by denoted x of function sine inverse the called is y x y sin relation the by determined x of function a is y if Functions ric Trigonomet Inverse of Properties and s Definition call Re 1 - -1 x if 0 y 2 π - or 1 x if π/2 y 0 : where x y csc if x 1 csc y -1 x if y π/2 or 1 x if π/2 y 0 : where x y sec if x 1 - sec y π y 0 : where x y cot if x 1 cot y π/2 y π/2 - : where x y tan if x 1 tan y π y 0 : where x cos y if x 1 cos y π/2 y π/2 - : where x y sin if x 1 sin y : s definition following the are these general, In      
DIFFERENTIATION FORMULA Derivative of Inverse Trigonometric Function functions. ric trigonomet other the for formulas the derive can we manner similar In x - 1 1 dx x sin d x sin y but x - 1 1 dx dy x - 1 y sin - 1 y cos : identity the from y cos 1 dx dy or dy dx y cos : y to respect with ting ifferentia D 2 y 2 - where x y sin function ric trigonomet inverse of definition the use we , x sin y of derivative the finding In 2 1 - 1 - 2 2 2 -1 dx du u - 1 1 u sin dx d Therefore 2 1 -
DIFFERENTIATION FORMULA Derivative of Inverse Trigonometric Function dx du 1 u u 1 u csc dx d 6. dx du 1 u u 1 u sec dx d 5. dx du u 1 1 u cot dx d 4. dx du u 1 1 u tan dx d 3. dx du u 1 1 u cos dx d 2. dx du u 1 1 u sin dx d 1. : functions ric trigonomet inverse for formulas ation Differenti 2 1 2 1 2 1 2 1 2 1 2 1
A. Find the derivative of each of the following functions and simplify the result: EXAMPL E : 3 1 x sin x f . 1 2 2 3 3x x 1 1 (x) f' 6 6 6 2 x 1 x 1 x 1 3x x f' x 3 cos x f . 2 1 2 2 2 9x 1 9x 1 9x 1 3 x f' 3 3x 1 1 x f' 2 6 6 2 x 1 x 1 3x x f' 2 2 9x 1 9x 1 3 x f' 6 2 x 1 3x x f' 2 9x 1 3 x f'
2 1 x 2 sec y . 3 4x 1 2x 2x 1 y' 2 2 2 1 4x x 2 y' 4 x cos 2 y . 4 1 x 2 1 x 1 1 2 ' y 2 x

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