y=arcsin(-4x) Evaluate (dy)/(dx) at x=-(1)/(6). Use an exact expression.

Apply Chain Rule: To find the derivative of y=arcsin(-4x), we use the chain rule. The derivative of arcsin(u) with respect to u is 1/sqrt(1-u^2), so we need to multiply that by the derivative of -4x with respect to x, which is -4. dy/dx = (1/sqrt(1-(-4x)^2)) * (-4)Simplify Derivative: Simplify the expression inside the square root and the derivative becomes: dy/dx = (-4)/(sqrt(1-16x^2))Evaluate at x=-(1/6): Now we evaluate the derivative at x=-(1/6). dy/dx at x=-(1/6) = (-4)/(sqrt(1-16(-1/6)^2))Simplify Inside Square Root: Simplify the expression inside the square root: dy/dx at x=-(1/6) = (-4)/(sqrt(1-16(1/36)))Further Simplify Expression: Further simplify the expression: dy/dx at x=-(1/6) = (-4)/(sqrt(1-(16/36)))Simplify Fraction: Simplify the fraction 16/36 to 4/9: dy/dx at x=-(1/6) = (-4)/(sqrt(1-(4/9)))Subtract from 1: Subtract the fraction from 1: dy/dx at x=-(1/6) = (-4)/(sqrt((9/9)-(4/9)))Simplify Subtraction: Simplify the subtraction inside the square root: dy/dx at x=-(1/6) = (-4)/(sqrt(5/9))Take Square Root: Take the square root of the fraction: dy/dx at x=-(1/6) = (-4)/(sqrt(5)/3)Multiply by 3: Multiply the numerator and denominator by 3 to get rid of the fraction in the denominator: dy/dx at x=-(1/6) = (-4*3)/(sqrt(5))Simplify the multiplication: dy/dx at x=-(1/6) = -12/sqrt(5)

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Find derivatives of inverse trigonometric functions

Full solution

y=\arcsin (-4 x)

\newline

Evaluate

\frac{d y}{d x}

x=-\frac{1}{6}

\newline

Use an exact expression.

Apply Chain Rule: To find the derivative of $y=\arcsin(-4x)$ , we use the chain rule. The derivative of $\arcsin(u)$ with respect to $u$ is $\frac{1}{\sqrt{1-u^2}}$ , so we need to multiply that by the derivative of $-4x$ with respect to $x$ , which is $-4$ . $\frac{dy}{dx} = \left(\frac{1}{\sqrt{1-(-4x)^2}}\right) \cdot (-4)$
Simplify Derivative: Simplify the expression inside the square root and the derivative becomes: $\frac{dy}{dx} = \frac{-4}{\sqrt{1-16x^2}}$
Evaluate at $x=-(\frac{1}{6})$ : Now we evaluate the derivative at $x=-(\frac{1}{6})$ .
$\frac{dy}{dx}$ at $x=-(\frac{1}{6})$ = $\frac{-4}{\sqrt{1-16(-\frac{1}{6})^2}}$
Simplify Inside Square Root: Simplify the expression inside the square root: $\frac{dy}{dx}$ at $x=-\left(\frac{1}{6}\right) = \frac{-4}{\sqrt{1-16\left(\frac{1}{36}\right)}}$
Further Simplify Expression: Further simplify the expression: $\frac{dy}{dx}$ at $x=-\left(\frac{1}{6}\right) = \frac{-4}{\sqrt{1-\left(\frac{16}{36}\right)}}$
Simplify Fraction: Simplify the fraction $\frac{16}{36}$ to $\frac{4}{9}$ : $\newline$ $\frac{dy}{dx}$ at $x=-\left(\frac{1}{6}\right) = \frac{-4}{\sqrt{1-\left(\frac{4}{9}\right)}}$
Subtract from $1$ : Subtract the fraction from $1$ : $\frac{dy}{dx}$ at $x=-\left(\frac{1}{6}\right) = \frac{-4}{\sqrt{\left(\frac{9}{9}\right)-\left(\frac{4}{9}\right)}}$
Simplify Subtraction: Simplify the subtraction inside the square root: $\frac{dy}{dx}$ at $x=-(\frac{1}{6}) = \frac{-4}{\sqrt{\frac{5}{9}}}$
Take Square Root: Take the square root of the fraction: $\frac{dy}{dx}$ at $x=-\left(\frac{1}{6}\right) = \frac{-4}{\sqrt{5}/3}$
Multiply by $3$ : Multiply the numerator and denominator by $3$ to get rid of the fraction in the denominator: $\newline$ $\frac{dy}{dx}$ at $x=-(\frac{1}{6})$ = $\frac{-4\times 3}{\sqrt{5}}$
Multiply by $3$ : Multiply the numerator and denominator by $3$ to get rid of the fraction in the denominator: $\newline$ $\frac{dy}{dx}$ at $x=-(\frac{1}{6})$ = $\frac{-4\times 3}{\sqrt{5}}$ Simplify the multiplication: $\newline$ $\frac{dy}{dx}$ at $x=-(\frac{1}{6})$ = $\frac{-12}{\sqrt{5}}$