The Derivative of Inverse Cosine Function | Explained using Chain Rule and Trigonometric Identity

derivative of cos^-1

The derivative of the inverse cosine function, or cos^(-1)(x), can be found using the chain rule of differentiation

The derivative of the inverse cosine function, or cos^(-1)(x), can be found using the chain rule of differentiation. The chain rule states that if we have a composite function, such as f(g(x)), then the derivative can be found by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

In this case, let’s consider the inverse cosine function as the outer function, denoted as y = cos^(-1)(x). The inner function is x = cos(y).

To find dy/dx, we can differentiate both sides of the equation x = cos(y) with respect to x:

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

The left-hand side, d/dx (x), is simply 1.

On the right-hand side, we need to use the chain rule. The derivative of cos(y) with respect to y is given by -sin(y). However, since we want dy/dx and not dy/dy, we need to divide the derivative by dy/dx. This means we have:

1 = -sin(y) * dy/dx.

Now, we need to solve for dy/dx. Since x = cos(y), we can substitute cos(y) for x:

1 = -sin(y) * dy/dx.

1 = -sin(cos^(-1)(x)) * dy/dx.

Finally, we isolate dy/dx by dividing both sides of the equation by -sin(cos^(-1)(x)):

dy/dx = 1 / (-sin(cos^(-1)(x))).

Simplifying further, we can use the fundamental trigonometric identity sin^2(y) + cos^2(y) = 1:

dy/dx = 1 / (-sin(cos^(-1)(x))) = 1 / (-√(1-x^2)).

Therefore, the derivative of cos^(-1)(x) with respect to x is 1 / (-√(1-x^2)).

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