Find derivatives of inverse trigonometric functions

Full solution

y=\arcsin (-4 x)

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Evaluate

\frac{d y}{d x}

x=-\frac{1}{6}

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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 $x=-(\frac{1}{6})$ = $\frac{-12}{\sqrt{5}}$