6.5 Derivatives of Inverse Trigonometric Functions

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6.5 Derivatives of Inverse Trigonometric
Functions Approach to differentiate an inverse
trigonometric function, we reduce the expression
to trigonometric functions and use the rule of
implicit differentiation. Example
Differentiate the arcsine function Solution
This same equality can be rewritten as We
need to find dy/dx, which can be done
implicitly Next, we want to write this
derivative as a function of x, not y
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Example (cntd) To reduce the derivative to
the function of x, we use the fact that sin yx,
and the trigonometric identity that gives We
use the fact that the range of the arcsine
function is restricted to Since the cosine
function takes only positive values in this
interval, the positive sign must be
chosen Using this equality, we write the
derivative in the final form
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Exercise Differentiate the arccosine
function Example Differentiate the
arctangent function Solution Differentiate
implicitly Apply some trigonometry to
write the result as a function of x
Finally
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Inverse trigonometric rules
Generalized inverse trigonometric rules (using
chain rule)
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Exercises Differentiate
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Homework Section 6.5 1,7,19,25,27,31,37,41.
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