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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