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

Evaluate $\frac{d}{d x}[\arccos (-2 x)]$ at $x=\frac{1}{4}$ . $\newline$ Use an exact expression.

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Find derivatives of inverse trigonometric functions

Full solution

Q. Evaluate

\frac{d}{d x}[\arccos (-2 x)]

at

x=\frac{1}{4}

.

\newline

Use an exact expression.

Find derivative of $f(x)$ : step_1: Find the derivative of $f(x) = \text{arccos}(-2x)$ . The derivative of $\text{arccos}(x)$ is $-\frac{1}{\sqrt{1-x^2}}$ , so by the chain rule, the derivative of $\text{arccos}(-2x)$ is $-\frac{1}{\sqrt{1-(-2x)^2}} \cdot \frac{d}{dx}(-2x)$ . $f'(x) = -\frac{1}{\sqrt{1-(-2x)^2}} \cdot (-2) = \frac{2}{\sqrt{1-4x^2}}$ .
Evaluate at $x = \frac{1}{4}$ : step_2: Evaluate the derivative at $x = \frac{1}{4}$ . $f'(\frac{1}{4}) = \frac{2}{\sqrt{1-4*(\frac{1}{4})^2}} = \frac{2}{\sqrt{1-1}} = \frac{2}{\sqrt{0}}$ .

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