Derivative of Inverse Trigonometric functions

The Inverse Trigonometric functions are also called as arcus functions, cyclometric functions or anti-trigonometric functions. These functions are used to obtain angle for a given trigonometric value. Inverse trigonometric functions have various application in engineering, geometry, navigation etc.

Representation of functions:

Generally, the inverse trigonometric function are represented by adding arc in prefix for a trigonometric function, or by adding the power of -1, such as:

Inverse of sin x = arcsin(x) or

sin1x

Let us now find the derivative of Inverse trigonometric function

Example: Find the derivative of a function

y=sin1x
.

Solution:Given

y=sin1x
…………(i)

x=siny

Differentiating the above equation w.r.t. x, we have:

dydx=1cosy

Putting the value of y form (i), we get

dydx=1cosy=1cos(sin1x)
………..(ii)

From equation (ii), we can see that the value of cos y cannot be equal to 0, as the function would become undefined.

sin1xπ2,π2

i.e.

x1,1

From (i) we have

y=sin1x

siny=sin(sin1x)

Using property of trigonometric function,

cos2y=1sin2y=1(sin(sin1x))2=1x2

cosy=1x2
…………(iii)

Now putting the value of (iii) in (ii), we have

dydx=11x2

Therefore, the Derivative of Inverse sine function is

ddx(sin1x)=11x2

Derivatives of Inverse trigonometric functions

Function (
dydx
)
arcsin x
11x2
arccos x
11x2
arctan x
11+x2
arccot x
11+x2
arcsec x
1|x|x21
arccsc x
1|x|x21
Example:Find the derivative of a function 2 arcsin x – 5 arccsc x.

Solution:Given

2arcsinx5arccscx

dydx=21x2+5xx21

Further we can factorize the given expression.

Example:Find the derivative of a function

sin1(1x21+x2)
.

Solution:Given y =

sin1(1x21+x2)

dydx=11(1x21+x2)2×ddx(1x21+x2)

dydx=1(1+x2)2(1x2)2(1+x2)2×ddx(1x21+x2)

dydx=1+x2(1+x4+2x2)(1+x42x2)×((1+x2)(2x)(1x2)(2x))(1+x2)2)

dydx=1+x24x2×((2x2x32x+2x3)(1+x2)2)

dydx=1+x22x×(4x(1+x2)2)

dydx=21+x2

Video Lesson on Trigonometry

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Quiz on Derivative of Inverse Trigonometric functions

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  1. Nice explain 😃 thank you so much for all byjus Teachers 🙏🖤