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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:

xy Arcsin

.sin xy

.cos

1 1cos sin

ydx

dy

dx

dyyx

dx

dy

dx

d

2

Example (cntd): To reduce the derivative to the function of x, we use the fact that sin y=x, 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:

.22

y

,1cossin 22 yy

.1sin1cos 22 xyy

.1cos 2xy

21

1

xdx

dy

3

Exercise: Differentiate the arccosine function:

Example: Differentiate the arctangent functionSolution: Differentiate implicitly:

Apply some trigonometry to write the result as a function of x:

Finally:

xy Arccos

xyxy tan Arctan

.sec

1 1sec tan

22

ydx

dy

dx

dyyx

dx

dy

dx

d

.tan1cos

sincos

cos

1sec 2

2

22

22 y

y

yy

yy

.1

1

tan1

122 xydx

dy

4

dx

du

uu

dx

du

uu

dx

du

uu

2

2

2

1

1tanArc

1

1cosArc

1

1sinArc

dx

d

dx

d

dx

d

Generalized inverse trigonometric rules (using chain rule)

Inverse trigonometric rules

2

2

2

xx

dx

dx

xdx

dx

xdx

d

1

1tanArc

1

1cosArc

1

1sinArc

5

Exercises:Differentiate

24tanArc3 xy

)1(Arccos 2xy

21Arctan xy

2Arcsin xxy

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HomeworkSection 6.5: 1,7,19,25,27,31,37,41.