First Derivative
Definition
The first derivative represents the rate at which a function is changing at any given point. It measures the slope of the tangent line to the graph of a function at that point.
Analogy
Imagine you're driving a car on a curvy road. The first derivative is like your speedometer, telling you how fast you're going at any moment. If the first derivative is positive, it means you're accelerating; if it's negative, it means you're decelerating.
Related terms
Second Derivative: The second derivative represents the rate at which the first derivative is changing. It helps determine whether a function is concave up or concave down.
Critical Points: Critical points are locations where either the first derivative equals zero or does not exist. They can indicate maximums, minimums, or inflection points on a graph.
Rate of Change: Rate of change refers to how quickly one quantity changes with respect to another quantity. In calculus, it often involves finding derivatives to measure this rate accurately.
"First Derivative" appears in:
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Practice Questions (20+)
You are using calculus to model a population of pandas. In your model the first derivative represents the rate of change of the population with respect to time. What are the units of the first derivative?
A tank is being filled with gas, and the first derivative represents the rate at which the liters of gas are increasing. What are the units and what is the sign of the first derivative?
The number of people in a livestream is decreasing at a variable rate, and the first derivative represents the rate at which people are leaving the room with respect to time. What are the units of the second derivative?
The temperature of a room is decreasing at a constant rate, and the first derivative represents the rate at which the temperature is decreasing. What are the units and what is the sign of the second derivative.
While driving his truck, David uses up gasoline. If g(x) is the amount of gasoline left in his truck after driving x miles, what is the sign of the first derivative, g'(x)?
Water is leaking from a tank. The amount of water in the tank at a time, t, is modeled by the function w(t). Interpret the first derivative w'(t) in context.
What is the first derivative of a function if its second derivative is equal to zero at a particular point?
If the first derivative of a function is positive, what can be said about the original function?
When the first derivative of a function changes sign from positive to negative, what can be inferred about the behavior of the function?
If the first derivative of a function is zero for an interval, what can be said about the behavior of the function on that interval?
If the first derivative of a function is negative, what can be said about the original function?
If the first derivative of a function is always positive, what can be said about the original function?
When the first derivative of a function changes sign from negative to positive, what can be inferred about the behavior of the function?
What does it mean if the first derivative of a function is negative at a specific point?
Which of the following is true about a function with a negative first derivative and a positive second derivative?
If a function has a positive first derivative and a negative second derivative, what can be said about the function's behavior?
What is the relationship between the sign of the first derivative and the increasing or decreasing behavior of a function?
What key feature of a function can be determined by analyzing the sign of the first derivative at a critical point?
The second derivative of the human population P with respect to time is proportional to the first derivative of P with respect to time. Which of the following differential equations represents the situation?
If the graph of the first derivative $f(x)$ of an integrally-defined function $h(x)$ is below the x-axis on an interval $(e, f)$, what can be concluded about the behavior of $h(x)$ on that interval?
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