The derivative of arctan returns an algebraic expression. We can use this expression to find the derivative of inverse trigonometric functions. In our discussion, we’ll understand how to differentiate arctan and use the new derivative rule to differentiate more complex functions. Make sure to have your notes handy! Calculate the derivative of the function (y = arccos x arctan x ) at (x = 0. ) Example 9 Using the chain rule, derive the formula for the derivative of the inverse sine function.

What is the derivative of #arctan (1/x)#?

1 Answer

Explanation:

We seek:

# d/dx arctan(1/x) #

Derivative Of Arctan(x/a)

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# d/dx tanx=1/(1+x^2)#

In conjunction with the power rule and the chain rule we get:

# d/dx arctan(1/x) = 1/(1+(1/x)^2) d/dx (1/x) #

# ' ' = 1/(1+1/x^2) (-1/x^2) #

# ' ' = -1/((1+1/x^2)x^2) #

# ' ' = -1/(x^2+1) #

Observation:

Derivative Of Arctan Formula

The astute reader will notice that:

# d/dx arctan(1/x) = -d/dx arctan x #

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# arctan(1/x) = -arctan x + C => arctan(1/x)+arctan x = C #

Although this result may look like an error, it is in fact correct, and a standard trigonometric result:

What is the derivative of #arctan (1/x)#?

1 Answer

Explanation:

Derivative Of Arctan(x/a)

Derivative Of Arctan Formula

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