Difference between Gradient, Rate of Change and Derivative (Single Variable Calculus)

Question

I was just wondering what the differences between gradient, rate of change and derivative are in terms of single variable calculus. As in can we use “gradient", “rate of change” and "derivative" interchangeably when talking about a function? Thanks in advance.

Answer 1

There are differences in meaning. "Derivative" is the broadest term. It's a certain limit. "Rate of change" is more specialized. It's the derivative with respect to time. I've never heard "gradient" used with a single-variable function, but I suppose it's technically correct. But it's still a more narrow term than "derivative", because functions $f:\mathbb{R}^n \to \mathbb{R}^m,$ with $m>1$, have derivatives but not gradients.

I think using gradient in single variable calculus would just be confusing.