Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
CHAPTER 2 DIFFERENTIATION 2.1 Differentiation of hyperbolic functions 2.2 Differentiation of inverse trigonometric functions 2.3 Differentiation of inverse hyperbolic functions *Recall: Methods of differentiation - Chain rule - Product differentiation - Quotient differentiation - Implicit differentiation 1 2.1 Differentiation of Hyperbolic Functions Recall: Definition: e x e-x cosh x 2 e x e-x sinh x 2 sinh x e x e-x tanh x x cosh x e e-x 1 cosh x coth x tanh x sinh x cosech x = sech x 1 sinh x 1 cosh x 2 Derivatives of hyperbolic functions Example 2.1: Find the derivatives of (a) sinh x (b) cosh x (c) tanh x Solution: d e x e x (a) sinh x dx dx 2 d 1 2 ( e x e x ) cosh x e x e x d d (b) cosh x dx dx 2 1 2 ( e x e x ) sinh x 3 e x e x d d (c) tanh x dx dx e x e x Using quotient diff: e x e x e x e x e x e x e x e x x x 2 e e e 2 x e 2 x 2 e 2 x e 2 x 2 e x e 4 e x e x 2 x 2 2 x x e e cosh x 1 2 2 sech 2 x Using the same methods, we can obtain the derivatives of the other hyperbolic functions and these gives us the standard derivatives. 4 Standard Derivatives y f ( x) dy f ( x) dx cosh x sinh x sinh x cosh x tanh x sech 2 x sech x cosech x coth x sech x tanh x cosech x coth x cosech 2 x 5 Example 2.2: 1.Find the derivatives of the following functions: a) b) c) 2. Find the derivatives of the following functions: (a) y cosh 3x (b) r sinh(2t 2 1) (c) g ( x) ( x 1)3 sech 2 x (d) y tanh(ln x) 3.(Implicit differentiation) Find dy from the following expressions: dx (a) x y 2 sinh 4 x cosh y (b) y tanh ( x y ) 6 2.2 Differentiation Involving Inverse Trigonometric Functions Recall: Definition of inverse trigonometric functions Function Domain sin 1 x 1 x 1 cos1 x 1 x 1 tan 1 x x sec1 x cot 1 x cosec1 x x 1 Range y 2 2 0 y y 2 0 y 2 2 2 y 0 y x x 1 2 y 00 y 2 7 Derivatives of Inverse Trigonometric Functions Standard Derivatives: d 1 1 1. (sin x) dx 1 x2 d 1 1 (cos x) 2. dx 1 x2 d 3. dx d 4. dx d 5. dx d 6. dx (tan (cot (sec 1 1 1 x) x) 1 1 x 2 1 1 x 2 1 x) x (csc 1 x 2 1 1 x) x x 2 1 8 2.2.1 Derivatives of y sin 1 x . (proof) Recall: y sin 1 x x sin y for x [1, 1] and y [ 2 , 2] . Because the sine function is differentiable on [ 2 , 2] , the inverse function is also differentiable. To find its derivative we proceed implicitly: Given sin y x . Differentiating w.r.t. x: d d (sin y ) ( x) dx dx cos y Since 2 y 2 dy 1 dx dy 1 dx cos y , cos y 0 , so dy 1 1 1 dx cos y 1 sin 2 y 1 x2 9 Example 2.3: 1. Differentiate each of the following functions. (a) f ( x) tan1 x (b) g (t ) sin 1(1 t ) (c) h( x) sec1 e2 x 2. Find the derivative of: (a) y (tan1 x 2 )4 (b) f ( x) ln(sin1 4 x) 3. Find the derivative of 4. Find the derivative y tan 1 (tan(3t 2 1)) . dy if dx (a) x tan1 y x 2 y (b) sin 1 ( xy) 2 cos1 y 10 Summary If u is a differentiable function of x, then d 1 du (sin 1 u ) 1. dx 1 u 2 dx d 1 du (cos1 u ) 2. dx 1 u 2 dx d 1 du 1 3. dx (tan u ) 1 u 2 dx d 1 du 1 (cot u ) 4. dx 1 u 2 dx d 1 du 1 (sec u ) 5. dx u u 2 1 dx d 1 du 1 (csc u ) 6. dx u u 2 1 dx 11 2.3 Derivatives of Inverse hyperbolic Functions Recall: Inverse Hyperbolic Functions Function Domain Range y sinh 1 x ( , ) ( , ) y cosh 1x [ 1, ) [ 0 , ) y tanh 1 x (1, 1) ( , ) y coth 1 x ( , 1) (1, ) ( , 0 ) ( 0 , ) y = sech1 x (0, 1] [ 0 , ) y = cosech1x ( , 0 ) ( 0 , ) ( , 0 ) ( 0 , ) Function Logarithmic form y sinh 1 x ln x x 2 1 y cosh 1x y tanh 1 x ln x x 1 2 1 1 x ln ; x 1 2 1 x 12 2.3.1 Proof: ( d 1 ) (sinh 1 x) 2 dx 1 x Recall: y sinh 1 x x sinh y To find its derivative we proceed implicitly: Given x sinh y . Differentiating w.r.t. x: d d ( x) (sinh y ) dx dx dy 1 cosh y dx dy 1 dx cosh y Since y , cosh y 0 , so using the identity cosh 2 y sinh 2 y 1: dy 1 1 1 dx cosh y 1 sinh2 y 1 x 2 d 1 (sinh 1 x) dx 1 x2 13 Other ways to obtain the derivatives are: (a) y sinh 1 x x sinh y then e y e y dy x . Hence, find . 2 dx (b) y sinh 1 x = ln x x 2 1 . Hence, find dy . dx 14 Standard Derivatives Function, y Derivatives, sinh 1 x dy dx 1 x2 1 cosh 1 x 1 x 1 2 ; x 1 tanh 1 x 1 ; x 1 2 1 x coth 1 x 1 ; x 1 2 1 x sech 1 x cosech1x 1 x 1 x 2 ; 0 x 1 1 x 1 x 2 ; x0 15 Generalised Form y f (u ); u g ( x) sinh 1 u dy dy du dx du dx 1 du u 2 1 dx cosh 1u du ; u 1 2 u 1 dx tanh 1 u 1 du ; u 1 2 1 u dx coth 1u 1 du ; u 1 2 1 u dx sech 1u cosech1u 1 1 du ; 0 u 1 2 dx u 1 u 1 du ; u0 2 dx u 1 u 16 Example 2.4: Find the derivatives of (a) y sinh 1 (1 3x) 1 (b) y cosh 1 x (c) y e xsech 1x (d) y sinh 1 (tan 3x) 1 2 tanh t (e) f (t ) 1 sec t (f) y coth 1 u (g) y cos 4 x cosh 1 4 x (h) y3 sinh 1 xy 0 17