Question

discuss the validity of the following statement. if the statement is always true, explain why. if not, give a counterexample.
the function ( h(x) = 5e^{x} ) is an antiderivative of itself.
if a function is an antiderivative of itself, then it is also a derivative of itself. since ( h(x)=square ), the statement is ( square )

Explanation:

Step1: Find the derivative of \(h(x)\)

Use the derivative formula \((e^x)^\prime = e^x\) and the constant - multiple rule \((cf(x))^\prime = cf^\prime(x)\).
For \(h(x)=5e^x\), by the constant - multiple rule \(h^\prime(x)=(5e^x)^\prime\).
Since \((e^x)^\prime = e^x\), then \(h^\prime(x)=5(e^x)^\prime\).
So \(h^\prime(x)=5e^x\).

Step2: Check the relationship between \(h(x)\) and \(h^\prime(x)\)

If \(h(x)\) is an antiderivative of itself, then \(h^\prime(x)\) should equal \(h(x)\).
We found \(h^\prime(x) = 5e^x\) and \(h(x)=5e^x\).

Answer:

\(h^\prime(x)=5e^x\), the statement is True.