Derivatives of Inverse Trig Functions

In a Nut Shell: You must know the derivatives of basic functions appearing in calculus.

Derivatives of the inverse trig functions are perhaps less used. The table below lists

each derivative of the inverse trig functions.

Derivatives of Inverse Trigonometric Functions are:

d/dx[sin^˗1 x ] = 1/(1 ˗ x²)^1/2, d/dx[cos^˗1 x] = - 1/(1 ˗ x²)^1/2, d/dx[tan^˗1 x] = 1/(1+x²)

d/dx[cot^˗1 x ] = -1/(1+x²) , d/dx[sec^˗1 x] = 1/[x(x²˗1)^1/2], d/dx[csc^˗1 x] = -1/[x(x²˗1)^1/2]

Strategy to calcuate derivatives of inverse trig functions

You can use the following strategy to find derivatives of inverse trig functions instead of

memorizing them.

Define a new variable, w, where w = tan^-1 (x)

Then tan w = x. Now take the derivative with respect to x on both sides.

sec² w dw/dx = 1 so the derivative of the inverse function is

dw/dx = cos²w

Since tan w = x Construct the following diagram to find cos w.

From the above diagram cos w = √[ 1/ (1 + x² ) ]

So the derivative of the inverse tangent function is 1 / (1 + x² )

Click here for an example.

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