find the derivative of sqrt(1 - x^2)

Описание к видео find the derivative of sqrt(1 - x^2)

Problem:
Find the derivative of √(1 - x^2).

Solution:

Let f(x) = √(1 - x^2).

We can rewrite the function as:
f(x) = (1 - x^2)^(1/2)

Now, use the chain rule to differentiate.

Differentiate the outer function:
The derivative of u^(1/2) is (1/2) * u^(-1/2), where u = 1 - x^2.
So, the outer derivative is:
(1/2) * (1 - x^2)^(-1/2)

Differentiate the inner function:
The derivative of 1 - x^2 is -2x.

Now, multiply the outer and inner derivatives together:
f'(x) = (1/2) * (1 - x^2)^(-1/2) * (-2x)

Simplify:
f'(x) = -x / √(1 - x^2)

Answer:
The derivative of √(1 - x^2) is -x / √(1 - x^2).

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