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

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

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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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Calculus

Find derivatives of inverse trigonometric functions

Full solution

Q. Evaluate

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

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. $\newline$ $f'(4) = \frac{1}{\sqrt{1-\frac{16}{36}}} \cdot \left(\frac{1}{6}\right)$ . $\newline$ $f'(4) = \frac{1}{\sqrt{1-\frac{4}{9}}} \cdot \left(\frac{1}{6}\right)$ . $\newline$ $f'(4) = \frac{1}{\sqrt{\frac{5}{9}}} \cdot \left(\frac{1}{6}\right)$ . $\newline$ $f'(4) = \frac{1}{\sqrt{5}/3} \cdot \left(\frac{1}{6}\right)$ . $\newline$ $f'(4) = \frac{3}{6\sqrt{5}}$ . $\newline$ $f'(4) = \frac{1}{2\sqrt{5}}$ .