What’s the physical meaning of a derivative in calculus?

Answer 1

The answer to this question is going to change drastically depending on the pair of quantities you're computing the derivative over. Since Calculus studies the relationship between continuous quantities, there are a lot of things you can do with it. Here are a few examples of pairs of quantities that can be studied in a calculus setting:

1. The distance between two object vs time
2. Color of a distant star vs time
3. Temperature of a rod vs the distance along it

In each of these cases, the derivative tells you how fast one quantity changes as another changes at a constant rate. So, in the first example, the derivative is going to tell you how fast the distance changes as time passes. This is otherwise known as velocity (or if you take the absolute value, it's speed). In the second example, the derivative is going to tell you how fast the color of a star changes as time passes. The last example doesn't involve time at all, but we can still make sense of the derivative. We can imagine moving along the rod at a constant rate and seeing how fast the temperature changes as we do so.

Answer 2

Great question! Check out my video explanation.

Answer 3

What is a derivative?

In calculus, a derivative is an mathematical operation which tells us how a variable changes with respect to the the dependent variable. Therefore, the derivative of y with respect to x, dy/dx measures how the dependent variable y changes when the value of the independent variable x increases or decreases slightly.

Examples and applications of derivatives in calculus

The simplest example of a derivative in calculus will be to calculate the slope of a graph. Let us say y = f(x) is a curve. The slope of the curve at a given point x, which is essentially how the value of y changes with x, is measured by the quantity dy/dx.

Derivatives are very useful in the real world. Measuring velocity (dx/dt), forces (dv/dt), heat flow (dT/dt), finding solutions to complex equations, etc. Derivatives are also the foundation for understanding the laws of nature with the help of differential equations.

Answer 4

derivative, graphically = slope of the function

physically, = rate of change of speed = velocity

Answer 5

A derivative tells us how something is changing—in physics, it often means velocity (change in position over time). Mathematically, it’s the slope of the tangent line at a point. If you’re driving, your speedometer shows the derivative of your position: how fast your location is changing right now.