Find all possible functions with the given derivative.
-$$y ^ { \prime } = x ^ { 3 } - 6 x$$
A) $\frac { x ^ { 4 } } { 4 } - \frac { 6 x ^ { 2 } } { 2 } + C$
B) $\frac { x ^ { 3 } } { 3 } - \frac { 6 x ^ { 2 } } { 2 } + C$
C) $3 x ^ { 2 } - 6 + C$
D) $\frac { x ^ { 4 } } { 4 } + 6 x ^ { 2 } + C$

Question

Quizplus · Accepted Answer

To find the antiderivative of $y' = x^3 - 6x$, integrate each term separately. The antiderivative of $x^3$ is $\frac{x^4}{4}$ and the antiderivative of $-6x$ is $-3x^2$. Adding a constant $C$ gives the general solution $\frac{x^4}{4} - 3x^2 + C$, which matches option A.

Find All Possible Functions with the Given Derivative y′=x3−6xy ^ { \prime } = x ^ { 3 } - 6 xy′=x3−6x

Find All Possible Functions with the Given Derivative $y ^ { \prime } = x ^ { 3 } - 6 x$