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If f(x)=arccos(x), then what is the value of f^(')((sqrt3)/(2)) in simplest form?

If f(x)=arccos⁡(x) f(x)=\arccos (x) , then what is the value of f′(32) f^{\prime}\left(\frac{\sqrt{3}}{2}\right) in simplest form?

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Full solution

Q. If f(x)=arccos⁡(x) f(x)=\arccos (x) , then what is the value of f′(32) f^{\prime}\left(\frac{\sqrt{3}}{2}\right) in simplest form?
  1. Find Derivative: Determine the derivative of f(x)=arccos⁡(x)f(x) = \arccos(x).\newlineThe derivative of arccos⁡(x)\arccos(x) is −11−x2-\frac{1}{\sqrt{1-x^2}}.\newlineddx(f(x))=ddx(arccos⁡(x))\frac{d}{dx}(f(x)) = \frac{d}{dx}(\arccos(x))\newlinef′(x)=−11−x2f'(x) = -\frac{1}{\sqrt{1-x^2}}
  2. Substitute xx Value: Evaluate the derivative at x=32x = \frac{\sqrt{3}}{2}. We substitute x=32x = \frac{\sqrt{3}}{2} into the derivative formula. f′(32)=−11−(32)2f'(\frac{\sqrt{3}}{2}) = -\frac{1}{\sqrt{1-(\frac{\sqrt{3}}{2})^2}}
  3. Simplify Expression: Simplify the expression inside the square root.\newlineCalculate the square of 3/2\sqrt{3}/2 and subtract from 11.\newline1−(3/2)2=1−3/4=1/41 - (\sqrt{3}/2)^2 = 1 - 3/4 = 1/4
  4. Calculate Derivative Value: Calculate the value of the derivative at x=3/2x = \sqrt{3}/2. Now we have f′(3/2)=−1/1/4f'(\sqrt{3}/2) = -1/\sqrt{1/4} This simplifies to f′(3/2)=−1/1/4=−1/(1/2)=−2f'(\sqrt{3}/2) = -1/\sqrt{1/4} = -1/(1/2) = -2

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