What does it mean if the derivative of a function is a constant?

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I was doing a homework problem to find the derivative of an equation and got "7" as the answer. I was trying to think about what it means if a derivative is a constant like that, is it just that the function is linear?

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2 Answers

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You don't need integration to do this.

A helpful exercise might be to show that if a function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$, and $f'(x) = 0$ for all $x$ in $(a,b)$, then $f(x)$ is constant on $[a,b]$. Hint: Use the mean value theorem.

Your question can then be answered by considering the function $g(x) = f(x) - 7x$.

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$$\frac{dy}{dx}=7\implies dy=7 \,dx\implies \int dy=\int 7 \, dx\implies y=7x+C$$

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