Solve the equation. (dy)/(dx)=-(1)/(3)x^(3) Choose 1 answer: (A) y=-(x^(4))/(12)+C (B) y=-x^(2)+C (c) y=-(x^(4))/(12+C) (D) y=-x^(2+C)

Integrate with respect to x: To solve the differential equation (dy)/(dx)=-(1)/(3)x^(3), we need to integrate both sides with respect to x.Apply power rule for integration: The antiderivative of -(1)/(3)x^(3) with respect to x is found using the power rule for integration, which states that the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, where C is the constant of integration.Antiderivative of -(1)/(3)x^(3): Applying the power rule to -(1)/(3)x^(3), we get: ∫-(1)/(3)x^(3) dx = -(1)/(3) * ∫x^(3) dx = -(1)/(3) * (x^(3+1))/(3+1) + CSimplify the expression: Simplifying the expression, we get: -(1)/(3) * (x^(4))/(4) + C = -(x^(4))/(12) + CFinal antiderivative: Therefore, the antiderivative of -(1)/(3)x^(3) is y = -(x^(4))/(12) + C, which corresponds to answer choice (A).

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Solve the equation.

(dy)/(dx)=-(1)/(3)x^(3)
Choose 1 answer:
(A)
y=-(x^(4))/(12)+C
(B)
y=-x^(2)+C
(c)
y=-(x^(4))/(12+C)
(D)
y=-x^(2+C)

Solve the equation. $\newline$ $\frac{d y}{d x}=-\frac{1}{3} x^{3}$ $\newline$ Choose $1$ answer: $\newline$ (A) $y=-\frac{x^{4}}{12}+C$ $\newline$ (B) $y=-x^{2}+C$ $\newline$ (c) $y=-\frac{x^{4}}{12+C}$ $\newline$ (D) $y=-x^{2+C}$

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Calculus

Find derivatives of using multiple formulae

Full solution

Q. Solve the equation.

\newline

\frac{d y}{d x}=-\frac{1}{3} x^{3}

\newline

Choose

1

answer:

\newline

(A)

y=-\frac{x^{4}}{12}+C

\newline

(B)

y=-x^{2}+C

\newline

(c)

y=-\frac{x^{4}}{12+C}

\newline

(D)

y=-x^{2+C}

Integrate with respect to $x$ : To solve the differential equation $\frac{dy}{dx}=-\frac{1}{3}x^{3}$ , we need to integrate both sides with respect to $x$ .
Apply power rule for integration: The antiderivative of $-(1)/(3)x^{3}$ with respect to $x$ is found using the power rule for integration, which states that the integral of $x^n$ with respect to $x$ is $(x^{n+1})/(n+1) + C$ , where $C$ is the constant of integration.
Antiderivative of $-\frac{1}{3}x^{3}$ : Applying the power rule to $-\frac{1}{3}x^{3}$ , we get: $\newline$ $\int -\frac{1}{3}x^{3} \, dx = -\frac{1}{3} \times \int x^{3} \, dx = -\frac{1}{3} \times \frac{x^{3+1}}{3+1} + C$
Simplify the expression: Simplifying the expression, we get: $-\frac{1}{3} \times \frac{x^{4}}{4} + C = -\frac{x^{4}}{12} + C$
Final antiderivative: Therefore, the antiderivative of $-\frac{1}{3}x^{3}$ is $y = -\frac{x^{4}}{12} + C$ , which corresponds to answer choice $(A)$ .