y=arccos(-(x)/(3)) (dy)/(dx)=? Choose 1 answer: (A) (1)/(3sqrt(1-(x^(2))/(9))) (B) (-1)/(3sqrt(1-(x^(2))/(9))) (c) (1)/(3(1-(x^(2))/(9))) (ㄷ) (-1)/(3(1-(x^(2))/(9)))

Write Derivative of arccos(u): First, let's write down the derivative of arccos(u) with respect to u, which is -1/sqrt(1-u^2).Apply Chain Rule with u=-(x)/(3): Now, we need to apply the chain rule. Let u=-(x)/(3). The derivative of u with respect to x is -1/3.Calculate Derivative of y: So, the derivative of y with respect to x is the derivative of arccos(u) times the derivative of u with respect to x, which is (-1/sqrt(1-u^2)) * (-1/3).Substitute u back in: Substitute u back in to get (-1/sqrt(1-(-(x)/(3))^2)) * (-1/3).Simplify the Expression: Simplify the expression to get (-1/sqrt(1-(x^2)/(9))) * (-1/3).Final Result: The negatives cancel out, so we get (1)/(3sqrt(1-(x^2)/(9))).

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y=arccos(-(x)/(3))

(dy)/(dx)=?
Choose 1 answer:
(A)
(1)/(3sqrt(1-(x^(2))/(9)))
(B)
(-1)/(3sqrt(1-(x^(2))/(9)))
(c)
(1)/(3(1-(x^(2))/(9)))
(ㄷ)
(-1)/(3(1-(x^(2))/(9)))

$y=\arccos \left(-\frac{x}{3}\right)$ $\newline$ $\frac{d y}{d x}=?$ $\newline$ Choose $1$ answer: $\newline$ (A) $\frac{1}{3 \sqrt{1-\frac{x^{2}}{9}}}$ $\newline$ (B) $\frac{-1}{3 \sqrt{1-\frac{x^{2}}{9}}}$ $\newline$ (C) $\frac{1}{3\left(1-\frac{x^{2}}{9}\right)}$ $\newline$ (D) $\frac{-1}{3\left(1-\frac{x^{2}}{9}\right)}$

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Calculus

Find derivatives of using multiple formulae

Full solution

y=\arccos \left(-\frac{x}{3}\right)

\newline

\frac{d y}{d x}=?

\newline

Choose

1

answer:

\newline

(A)

\frac{1}{3 \sqrt{1-\frac{x^{2}}{9}}}

\newline

(B)

\frac{-1}{3 \sqrt{1-\frac{x^{2}}{9}}}

\newline

(C)

\frac{1}{3\left(1-\frac{x^{2}}{9}\right)}

\newline

(D)

\frac{-1}{3\left(1-\frac{x^{2}}{9}\right)}

Write Derivative of arccos(u): First, let's write down the derivative of $\text{arccos}(u)$ with respect to $u$ , which is $-\frac{1}{\sqrt{1-u^2}}$ .
Apply Chain Rule with $u=-\frac{x}{3}$ : Now, we need to apply the chain rule. Let $u=-\frac{x}{3}$ . The derivative of $u$ with respect to $x$ is $-\frac{1}{3}$ .
Calculate Derivative of y: So, the derivative of $y$ with respect to $x$ is the derivative of $\arccos(u)$ times the derivative of $u$ with respect to $x$ , which is $(-1/\sqrt{1-u^2}) \times (-1/3)$ .
Substitute $u$ back in: Substitute $u$ back in to get $(-1/\sqrt{1-(-\frac{x}{3})^2}) * (-\frac{1}{3})$ .
Simplify the Expression: Simplify the expression to get $(-\frac{1}{\sqrt{1-\frac{x^2}{9}}}$ * $-\frac{1}{3}$ .
Final Result: The negatives cancel out, so we get $(1)/(3\sqrt{1-(x^2)/(9)})$ .