JUST THE MATHS UNIT NUMBER 0.7 DIFFERENTIATION 7 (Inverse hyperbolic functions) by A.J.Hobson 0.7. Summary of results 0.7.2 The erivative of an inverse hyperbolic sine 0.7.3 The erivative of an inverse hyperbolic cosine 0.7.4 The erivative of an inverse hyperbolic tangent 0.7.5 Exercises 0.7.6 Answers to exercises
UNIT 0.7 - DIFFERENTIATION DERIVATIVES OF INVERSE HYPERBOLIC FUNCTIONS 0.7. SUMMARY OF RESULTS The erivatives of inverse trigonometric an inverse hyperbolic functions shoul be consiere as stanar results. They will be state here, first, before their proofs are iscusse.. 2. 3. where < sinh x <. where cosh x 0. where < tanh x <. x [sinh x] = x [cosh x] = + x 2, x2, x [tanh x] = x, 2 0.7.2 THE DERIVATIVE OF AN INVERSE HYPERBOLIC SINE We shall consier the formula an etermine an expression for y x. y = Sinh x Note: The use of the upper-case S in the formula is temporary; an the reason will be explaine shortly. The formula is equivalent to so we may say that x = sinh y, x y = cosh y + sin 2 y + x 2,
noting that cosh y is never negative. Thus, y x =. + x 2 Consier now the graph of the formula y = Sinh x which may be obtaine from the graph of y = sinh x by reversing the roles of x an y an rearranging the new axes into the usual positions. We obtain: y O x Observations (i) The variable x may lie anywhere in the interval < x <. (ii) For each value of x, the variable y has only one value. (iii) For each value of x, there is only one possible value of y, which is positive. x (iv) In this case (unlike the case of an inverse sine, in Unit 0.6) there is no nee to istinguish between a general value an a principal value of the inverse hyperbolic sine function. This is because there is only one value of both the function an its erivative. However, it is customary to enote the inverse function by sinh x, using a lower-case s rather than an upper-case S. Hence, x [sinh x] = + x 2. 2
0.7.3 THE DERIVATIVE OF AN INVERSE HYPERBOLIC COSINE We shall consier the formula an etermine an expression for y x. y = Cosh x Note: There is a special significance in using the upper-case C in the formula; the reason will be explaine shortly. The formula is equivalent to so we may say that x = cosh y, x y = sinh y ± cosh 2 y ± x 2. Thus, y x = x2. Consier now the graph of the formula y = Cosh x, which may be obtaine from the graph of y = cosh x by reversing the roles of x an y an rearranging the new axes into the usual positions. We obtain: y O x 3
Observations (i) The variable x must lie in the interval x. (ii) For each value of x in the interval x >, the variable y has two values one of which is positive an the other negative. (iii) For each value of x in the interval x >, there are only two possible values of y x, one of which is postive an the other negative. (iv) By restricting the iscussion to the part of the graph for which y 0, there will be only one value of y with one (positive) value of y for each value of x in the interval x. x The restricte part of the graph efines what is calle the principal value of the inverse cosine function an is enote by cosh x, using a lower-case c. Hence, x [cosh x] = x2. 0.7.4 THE DERIVATIVE OF AN INVERSE HYPERBOLIC TANGENT We shall consier the formula an etermine an expression for y x. y = Tanh x Note: The use of the upper-case T in the formula is temporary; an the reason will be explaine later. The formula is equivalent to so we may say that x = tanh y, x y = sech2 y tanh 2 y x 2. Thus, y x = x. 2 Consier now the graph of the formula y = Tanh x, 4
which may be obtaine from the graph of y = tanh x by reversing the roles of x an y an rearranging the new axes into the usual positions. We obtain: y O x Observations (i) The variable x must lie in the interval < x <. (ii) For each value of x in the interval < x <, the variable y has just one value. (iii) For each value of x in the interval < x <, there is only possible value of y x, which is positive. (iv) In this case (unlike the case of an inverse tangent in Unit 0.6) there is no nee to istinguish between a general value an a principal value of the inverse hyperbolic tangent function. This is because there is only one value of both the function an its erivative. However, it is customary to enote the inverse hyperbolic tangent by tan x using a lowercase t rather than an upper-case T. Hence, x [tanh x] = x 2. ILLUSTRATIONS. x [sin (tanh x)] = sech 2 x = sechx. tanh 2 x 5
2. x [cosh (5x 4)] = 5 (5x 4) 2, assuming that 5x 4 ; that is, x. 0.7.5 EXERCISES Obtain an expression for y x. in the following cases: y = cosh (4 + 3x), 2. assuming that 4 + 3x ; that is, x. y = cos (sinh x), 3. assuming that sinh x. y = tanh (cos x). 4. 5. assuming that x. y = cosh (x 3 ), y = tanh 2x + x. 2 6
0.7.6 ANSWERS TO EXERCISES. 3 (4 + 3x) 2. 2. cosh x. sinh 2 x 3. cosecx. 4. 3x 2 x6. 5. 2 x 2. 7