Understanding and Solving for the Derivative of the Inverse Cosine Function | Step-by-Step Guide

d/dx arccos

To find the derivative of the inverse cosine function, we can start by using the chain rule

To find the derivative of the inverse cosine function, we can start by using the chain rule. Let’s denote the inverse cosine function as y = arccos(x). Taking the derivative of both sides with respect to x, we have:

dy/dx = d/dx(arccos(x))

To apply the chain rule, we need to express arccos(x) in terms of a composition of functions. We know the derivative of cos(x) with respect to x is -sin(x). By rearranging the equation y = arccos(x), we can express it in terms of cos(y):

cos(y) = x

Now, differentiating both sides with respect to x using the chain rule, we get:

-d/dx(sin(y)) = 1

-d/dx(sin(y)) = 0

cos(y) * dy/dx = 0

Using the fact that cos(y) = x, we can substitute it back into the equation:

x * dy/dx = 0

Finally, solving for dy/dx, we find:

dy/dx = 0 / x = 0

Therefore, the derivative of arccos(x) with respect to x is 0.

In summary:

d/dx(arccos(x)) = 0

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