The Chain Rule for Finding the Derivative of the Inverse Cosine Function (Arccos(x)): A Step-by-Step Guide

Derivative of arccos(x)

To find the derivative of the inverse cosine function (arccos(x)), we can use the chain rule

To find the derivative of the inverse cosine function (arccos(x)), we can use the chain rule.

Let’s start by recalling the relationship between the cosine function and the inverse cosine function:

cos(arccos(x)) = x

Differentiating both sides with respect to x will help us find the derivative of the inverse cosine function:

d/dx [cos(arccos(x))] = d/dx [x]

Using the chain rule on the left side, we get:

-sin(arccos(x)) * d(arccos(x))/dx = 1

Now, let’s solve for d(arccos(x))/dx, which represents the derivative of arccos(x):

d(arccos(x))/dx = 1 / (-sin(arccos(x)))

Since sin^2(a) + cos^2(a) = 1 and cos(arccos(x)) = x, we can use these relationships to replace sin(arccos(x)):

d(arccos(x))/dx = 1 / (-sqrt(1 – x^2))

Therefore, the derivative of arccos(x) is equal to -1 / sqrt(1 – x^2).

To summarize:

d(arccos(x))/dx = -1 / sqrt(1 – x^2)

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