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The chain rule
applications |
Derivatives of the hyperbolic functions |
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Derivatives of inverse hyperbolic functions |
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Derivative of
the inverse function |
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Derivatives of the inverse trigonometric functions |
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Derivatives of the hyperbolic functions |
| We
use the derivative of the exponential function and the chain rule to
determine the derivative of the hyperbolic
sine and the hyperbolic cosine functions. |
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| We
find derivative of the
hyperbolic tangent and the hyperbolic cotangent functions applying
the quotient rule. |
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| Therefore,
derivatives of the hyperbolic functions are |
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Derivatives of inverse hyperbolic functions |
| We
use the derivative of the logarithmic function and the chain rule to
find the derivative of inverse hyperbolic functions. |
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| We
use the same method to find
derivatives of other inverse hyperbolic functions,
thus |
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Derivative of the inverse function |
| If
given a function y =
f (x) the derivative of
which y' (x) is
not 0
then, the derivative of the
inverse function |
| x
= f -1(y)
is |
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Example:
Find
the derivative x'
(y)
if the given function f
(x)
= x
+
ln x.
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| Solution: |
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Derivative of the inverse trigonometric functions |
| 1)
The derivative of the inverse of the sine function y =
sin -1x,
| x | < 1 and -p/2
< y < p/2 |
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if x = sin y,
then |
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| 2)
The derivative of the inverse of the cosine function y
= cos -1x =
p/2
-
sin
-1x,
| x | < 1, 0
< y < p |
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| 3)
The derivative of the
inverse of the tangent function y =
tan -1x,
-
oo
< x < oo
and -p/2
< y < p/2 |
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if x = tan y,
then |
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| 4)
The derivative of the
inverse of the cotangent function y =
cot -1x
= p/2
-
tan -1x, |
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| 5)
The derivative of the
inverse of the secant function y =
sec -1x
= cos -1(1/x), |
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| 6)
The derivative of the
inverse of the cosecant function y =
csc -1x
= sin -1(1/x), |
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| Therefore,
derivatives of the inverse trigonometric functions are |
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