You might not realize it, but every time you find an antiderivative of a function you are actually solving a differential equation! To clarify, suppose you are asked to compute the antiderivative of \(x^2\text{.}\)
However, this is precisely what it means to solve a differential equation, and since the antiderivative of \(x^2\) is \(\sfrac{x^3}{3} + c\text{,}\) we can conclude that the family of functions that satisfy the differential equation
\begin{equation*}
y'(x) = x^2
\end{equation*}
is given by the general solution:
\begin{equation*}
y(x) = \frac{x^3}{3} + c
\end{equation*}
This answer does not satisfy the differential equation. When you take the derivative of \(y = x^3 - 7\text{,}\) you get \(\dfrac{dy}{dx} = 3x^2\text{,}\) which is not equal to \(x^3 - 7\text{.}\)
\(\quad y = \frac14x^4 - 7x\)
This answer is missing the integration constant.
\(\quad y = \frac13x^4 - 7x + c\)
Double check your antiderivative calculation. The antiderivative of \(x^3\) is \(\dfrac14x^4\text{,}\) not \(\dfrac13x^4\text{.}\)
Saying \(y\) is the derivative of \(x^3 - 7\text{,}\) is the same as \(\left(x^3 - 7\right)' = y\)
\(\quad x^3 - 7\) is the derivative of \(y\)
Yes, this is exactly how the antiderivative of a function is defined.
\(\quad y\) is the antiderivative of \(x^3 - 7\)
Saying \(y\) is the antiderivative of \(x^3 - 7\) is the same thing as saying “when you take the derivative of \(y\) you should get \(x^3 - 7\)”
\(\quad x^3 - 7\) is the antiderivative of \(y\)
Saying \(x^3 - 7\) is the antiderivative of \(y\) is the same as saying “when you take the derivative of \(x^3 - 7\) you should get \(y\)”, which is not true.
