Derivative of arccos(x) using the chain rule | -1/sqrt(1-x^2)

d/dx arccos(x)

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

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

The chain rule states that if we have a composition of functions, h(x) = f(g(x)), then the derivative of h(x) with respect to x is given by the product of the derivative of the outer function f'(g(x)) and the derivative of the inner function g'(x). In this case, f(x) = arccos(x) and g(x) = x.

First, let’s find the derivative of the inner function g(x). Since g(x) = x, its derivative g'(x) is simply 1.

Now, let’s find the derivative of the outer function f'(x) = arccos'(x). To do this, we need to recall the derivative of the arccosine function.

The derivative of the arccosine function, denoted as arccos'(x) or d/dx (arccos(x)), is equal to -1/sqrt(1-x^2).
(The square root function is commonly denoted as √x.)

Combining the derivatives, we can use the chain rule:

d/dx (arccos(x)) = f'(g(x)) * g'(x)
= arccos'(x) * 1
= -1/sqrt(1-x^2)

Therefore, the derivative of the function f(x) = arccos(x) with respect to x is -1/sqrt(1-x^2).

Note that the derivative of the arccosine function can also be expressed as -1/√(1-x^2) or -1/(1-x^2)^(1/2), depending on the preferred form of notation.

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