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Evaluate
(d)/(dx)[arcsin((x)/(6))] at
x=4.
Use an exact expression.

Evaluate $\frac{d}{d x}\left[\arcsin \left(\frac{x}{6}\right)\right]$ at $x=4$ . $\newline$ Use an exact expression.

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Find derivatives of inverse trigonometric functions

Full solution

Q. Evaluate

\frac{d}{d x}\left[\arcsin \left(\frac{x}{6}\right)\right]

at

x=4

.

\newline

Use an exact expression.

Find Derivative of $f(x)$ : step_1: Find the derivative of $f(x) = \arcsin\left(\frac{x}{6}\right)$ . $\newline$ The derivative of $\arcsin(u)$ with respect to $x$ is $\frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx}$ . $\newline$ So, $f'(x) = \frac{1}{\sqrt{1-\left(\frac{x}{6}\right)^2}} \cdot \left(\frac{1}{6}\right)$ .
Simplify Derivative: step_2: Simplify the derivative. $f'(x) = \frac{1}{\sqrt{1-\frac{x^2}{36}}} \cdot \frac{1}{6}$ .
Evaluate at $x=4$ : step_3: Evaluate the derivative at $x=4$ . $f'(4) = \frac{1}{\sqrt{1-\frac{4^2}{36}}} \times \left(\frac{1}{6}\right)$ .
Calculate Value: step_4: Calculate the value. $\newline$ $f'(4) = \frac{1}{\sqrt{1-\frac{16}{36}}} \cdot \left(\frac{1}{6}\right)$ . $\newline$ $f'(4) = \frac{1}{\sqrt{1-\frac{4}{9}}} \cdot \left(\frac{1}{6}\right)$ . $\newline$ $f'(4) = \frac{1}{\sqrt{\frac{5}{9}}} \cdot \left(\frac{1}{6}\right)$ . $\newline$ $f'(4) = \frac{1}{\sqrt{5}/3} \cdot \left(\frac{1}{6}\right)$ . $\newline$ $f'(4) = \frac{3}{6\sqrt{5}}$ . $\newline$ $f'(4) = \frac{1}{2\sqrt{5}}$ .

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