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Assume that we have a function f(x) = 3x + 1, and I want to explain what actually derivative is. For this purpose, I constructed following sentence:

If you increment the value of variable x by 1, the result of the function is incremented by 3, and this is always constant because of constant slope.

In this sentence, I used the verb "increment" two times, and this sounds to me weird and not proper. Is there a more formal and proper way to express such a situation ? I think that the alternative verb for "increment" is "increase" or maybe "rise", but they are not appropriate candidates to replace second "increment" in the sentence.

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Increment means "increase to the next step on a fixed scale". Often it means "increase by one".

"Increase" would be a better verb to use, and the active "increases" the second time (instead of "is incremented")

If you increase the value of x by one, the value of the function increases by three.

I'm not sure about the mathematical clarity of the sentence. It seems that you are saying the the derivative is constant because the slope is constant. But how do you know the slope is constant (without finding the derivative)?

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  • I agree with you about "increases." By definition, the slope of a linear function is its derivative. – Katy Dec 5 '19 at 1:33
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In this sentence, I used the verb "increment" two times, and this sounds to me weird and not proper.

You can get away with this when talking about things with precision, such as technical or legal topics, or mathematics.

Increment isn't a common word in everyday conversation, so if you are trying to reach a non-technical audience, you want to step down from being formal, not be more formal. Raise might be a friendlier word.

If you raise the value of variable x by 1, the result of the function is raised by 3, and this is always constant because of constant slope.

Or you could speak in terms of add which is something everyone learns when they are young and is therefore pretty friendly and familiar:

If you add 1 to x, the result of the function goes up by 3, and this is always constant because of constant slope.

I'm not a calculus expert (I took one class) but I always thought of the derivative as "what do we do to find the next x?".

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