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Show that if two functions $f$ and $g$ have the same derivative on the same interval, then their difference, $f(x)-g(x)=$ constant on the interval. Hint: Define $D(x)=f(x)-g(x),$ show $D^{\prime}(x)=0$ on the interval and then deduce the result as a consequence of the first derivative test.

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 2

The First Derivative Test

Derivatives

Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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for the following problem, we wish to show that if two functions F and G have the same derivative on the same interval and their difference F of x minus G of X is constant on the interval. Um So we want to define dfx to be ffx minus g fx. Yeah, Yeah. And then we'll consider the fact that if we take the prime of X is equal to f prime of X minus G prime of X. But we're told that f prime of X and you have the same derivative on the same interval. So that's right here with equal zero. Because we ended things zero. This tells us that this term must be a constant, so it's going to be our final answer. And because it's constant, that means not that this change is constant, but F n G f minus G must be constant because it's derivative is zero.

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