The Chain Rule: Finding the Derivative of the Inverse Cosine Function (arccos(x))

d/dx arccos

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

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

The chain rule states that if we have a composition of two functions, f(g(x)), then the derivative of this composition is given by f'(g(x)) multiplied by g'(x).

In this case, our composition is arccos(x). Let’s denote the derivative of arccos(x) as d(arccos(x))/dx.

Let y = arccos(x), then we can rewrite it as cos(y) = x.

To find the derivative, we take the derivative of both sides with respect to x:

d(cos(y))/dx = d(x)/dx

Now, let’s apply the chain rule. The derivative of cos(y) with respect to y is -sin(y), and the derivative of x with respect to x is 1:

-sin(y) * dy/dx = 1

Now, we need to solve for dy/dx, the derivative of arccos(x) with respect to x:

dy/dx = -1/sin(y)

To solve for sin(y), we can use the right triangle definition of the cosine function. If we define a right triangle with angle y and adjacent side length x (since cos(y) = x), the opposite side can be found using the Pythagorean theorem:

opposite side = sqrt((adjacent side)^2 – (hypotenuse)^2)
opposite side = sqrt(x^2 – 1)

Since sin(y) is the ratio of the opposite side to the hypotenuse (sin(y) = opposite side / hypotenuse), we have:

sin(y) = sqrt(x^2 – 1) / 1
sin(y) = sqrt(x^2 – 1)

Plugging this back into the expression for dy/dx, we get:

dy/dx = -1/sqrt(x^2 – 1)

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

d(arccos(x))/dx = -1/sqrt(x^2 – 1)

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