The Derivative of the Inverse Cosine Function | A Step-by-Step Guide and Simplified Expression

Derivative of arccos(x)

To find the derivative of the inverse cosine function (arccosine), denoted as arccos(x) or cos^(-1)(x), we can use the chain rule

To find the derivative of the inverse cosine function (arccosine), denoted as arccos(x) or cos^(-1)(x), we can use the chain rule.

Let’s start by considering the cosine function, cos(x). The derivative of cos(x) with respect to x is given by -sin(x). Now, let’s find the derivative of the inverse cosine function, arccos(x), denoted as f(x).

We can define f(x) = arccos(x), such that x = cos(f(x)). Taking the cosine of both sides:

cos(x) = cos(arccos(x))

Using the inverse function property, we get:

x = arccos(x)

Differentiating both sides of this equation with respect to x using the chain rule, we obtain:

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

Rearranging this equation and solving for df/dx (the derivative of f(x) with respect to x), we have:

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

We can simplify this expression using the trigonometric identity sin^2(x) + cos^2(x) = 1. Since arccos(x) gives us an angle whose cosine is x, we can rewrite this identity as sin^2(arccos(x)) + cos^2(arccos(x)) = 1. Therefore, sin^2(arccos(x)) = 1 – x^2.

Substituting this into our derivative expression, we get:

df/dx = 1 / (-sin(arccos(x)))
= 1 / (-√(1 – x^2))

So, the derivative of arccos(x) is given by 1 / (-√(1 – x^2)).

It is important to note that the domain of the inverse cosine function is typically considered to be -1 ≤ x ≤ 1. If x is outside this interval, the derivative is undefined.

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