10.7. DIFFERENTIATION 7 (Inverse hyperbolic functions) A.J.Hobson

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1 JUST THE MATHS SLIDES NUMBER 0.7 DIFFERENTIATION 7 (Inverse hyperbolic functions) by A.J.Hobson 0.7. Summary of results The erivative of an inverse hyperbolic sine The erivative of an inverse hyperbolic cosine The erivative of an inverse hyperbolic tangent

2 UNIT DIFFERENTIATION 7 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, as follows: x [sinh x] = where < sinh x <. where cosh x 0. x [cosh x] = + x 2, x2, x [tanh x] = x 2 where < tanh x <.

3 0.7.2 THE DERIVATIVE OF AN INVERSE HYPERBOLIC SINE We shall consier the formula y = Sinh x an etermine an expression for y 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, x = sinh y; x y = cosh y + sin 2 y + x 2, noting that cosh y is never negative. Thus, y x =. + x 2 2

4 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. The variable x may lie anywhere in the interval < x <. 2. For each value of x, the variable y has only one value. 3. For each value of x, there is only one possible value of, which is positive. y x 3

5 4. There is no nee to istinguish between a general value an a principal value of the inverse hyperbolic sine function since 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. Hence, x [sinh x] = + x THE DERIVATIVE OF AN INVERSE HYPERBOLIC COSINE We shall consier the formula y = Cosh x an etermine an expression for y x. Note: There is a special significance in using the upper-case C in the formula; see later. The formula is equivalent to so, x = cosh y; x y = sinh y ± cosh 2 y ± x 2. 4

6 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 Observations. The variable x must lie in the interval x. 2. For each value of x in the interval x >, the variable y has two values one of which is positive an the other negative. 5

7 3. 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. 4. On the part of the graph for which y 0, there will be only one value of y with one (positive) value of y x for each value of x in the interval x. The restricte part of the graph efines the principal value of the inverse cosine function an is enote by cosh x using a lower-case c. Hence, x [cosh x] = x THE DERIVATIVE OF AN INVERSE HYPERBOLIC TANGENT We shall consier the formula y = Tanh x an etermine an expression for y x. Note: The use of the upper-case T in the formula is temporary; an the reason will be explaine shortly. The formula is equivalent to x = tanh y; 6

8 so, x y = sech2 y tanh 2 y x 2. Thus, y x = x 2. Consier now the graph of the formula y = Tanh x, 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 7

9 Observations. The variable x must lie in the interval < x <. 2. For each value of x in the interval < x <, the variable y has just one value. 3. For each value of x in the interval < x <, there is only possible value of y x, which is positive. 4. As with sinh x, there is no nee to istinguish between a general value an a principal value of the inverse hyperbolic tangent; but it is customary to enote it by tan x (lower-case t). Hence ILLUSTRATIONS. 2. x [sin (tanh x)] = x [cosh (5x 4)] = x [tanh x] = x 2. sech 2 x tanh 2 x = sechx. 5 (5x 4)2, assuming that 5x 4 ; that is, x. 8

UNIT NUMBER DIFFERENTIATION 7 (Inverse hyperbolic functions) A.J.Hobson

UNIT NUMBER DIFFERENTIATION 7 (Inverse hyperbolic functions) A.J.Hobson 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

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