746
Chapter 6 | Applications of Integration
⎛ ⎝ e
⎞ ⎠
x − e − x 2
d dx (sinh
x ) = d dx
⎡ ⎣ d
⎤ ⎦
e x )− d
e − x )
dx (
dx (
= 1 2
= 1 2 [ e x + e − x ] =cosh x . Similarly, ( d / dx )cosh x = sinh x . We summarize the differentiation formulas for the hyperbolic functions in the following table.
d dx
f ( x )
f ( x )
sinh x
cosh x
cosh x
sinh x
sech 2 x
tanh x
−csch 2 x
coth x
sech x
−sech x tanh x
csch x
−csch x coth x
Table 6.2 Derivatives of the Hyperbolic Functions
Let’s take a moment to compare the derivatives of the hyperbolic functions with the derivatives of the standard trigonometric functions. There are a lot of similarities, but differences as well. For example, the derivatives of the sine functions match: ( d / dx )sin x =cos x and ( d / dx )sinh x =cosh x . The derivatives of the cosine functions, however, differ in sign: ( d / dx )cos x =−sin x , but ( d / dx )cosh x = sinh x . As we continue our examination of the hyperbolic functions, we must be mindful of their similarities and differences to the standard trigonometric functions. These differentiation formulas for the hyperbolic functions lead directly to the following integral formulas. ∫ sinh udu = cosh u + C ∫ csch 2 udu = −coth u + C
∫ cosh udu = sinh u + C ∫ sech u tanh udu = −sech u + C ∫ sech 2 udu = tanh u + C ∫ csch u coth udu = −csch u + C
Example 6.47 Differentiating Hyperbolic Functions
Evaluate the following derivatives: a. d dx ⎛ ⎝ sinh ⎛ ⎝ x 2 ⎞ ⎠ ⎞ ⎠
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