Answer:
Explanation:
To find the derivative of (cos¹ (2x))(√x² - 4), we can use the product rule. The product rule states that the derivative of the product of two functions is equal to the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.
We can first find the derivative of the first function, cos¹ (2x). We can use the chain rule to find this. The chain rule states that the derivative of a composite function (two functions combined) is equal to the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is the inverse cosine function, and the inner function is 2x. The derivative of the inverse cosine function is 1/(cos(x))^2, and the derivative of 2x is 2. So, the derivative of cos¹ (2x) is (1/(cos(2x))^2)(2) = 2/(cos(2x))^2.
Next, we can find the derivative of the second function, √x² - 4. This is simply the derivative of the square root of a quadratic function, which is 1/(2√(x² - 4)).
Now, we can use the product rule to find the derivative of the entire expression. The derivative is (2/(cos(2x))^2)(1/(2√(x² - 4))) + (√x² - 4)(2/(cos(2x))^2) = 1/(cos(2x))^2 √(x² - 4) + 2√(x² - 4)/(cos(2x))^2.
This is the final derivative of (cos¹ (2x))(√x² - 4).