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Evaluate (d)/(dx)[arccos(-2x)] at x=(1)/(4). Use an exact expression.

Evaluate ddx[arccos⁡(−2x)] \frac{d}{d x}[\arccos (-2 x)] at x=14 x=\frac{1}{4} .\newlineUse an exact expression.

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Q. Evaluate ddx[arccos⁡(−2x)] \frac{d}{d x}[\arccos (-2 x)] at x=14 x=\frac{1}{4} .\newlineUse an exact expression.
  1. Find derivative of f(x)f(x): step_1: Find the derivative of f(x)=arccos(−2x)f(x) = \text{arccos}(-2x). The derivative of arccos(x)\text{arccos}(x) is −11−x2-\frac{1}{\sqrt{1-x^2}}, so by the chain rule, the derivative of arccos(−2x)\text{arccos}(-2x) is −11−(−2x)2⋅ddx(−2x)-\frac{1}{\sqrt{1-(-2x)^2}} \cdot \frac{d}{dx}(-2x). f′(x)=−11−(−2x)2⋅(−2)=21−4x2f'(x) = -\frac{1}{\sqrt{1-(-2x)^2}} \cdot (-2) = \frac{2}{\sqrt{1-4x^2}}.
  2. Evaluate at x=14x = \frac{1}{4}: step_2: Evaluate the derivative at x=14x = \frac{1}{4}.f′(14)=21−4∗(14)2=21−1=20f'(\frac{1}{4}) = \frac{2}{\sqrt{1-4*(\frac{1}{4})^2}} = \frac{2}{\sqrt{1-1}} = \frac{2}{\sqrt{0}}.

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