Derivative
Definition
A derivative represents the rate at which a function is changing at any given point. It measures how sensitive one quantity is to small changes in another quantity.
Analogy
Think of driving a car and looking at your speedometer. The speedometer tells you how fast your speed (quantity) is changing with respect to time (another quantity).
Related terms
Tangent Line: A tangent line represents a straight line that touches a curve at only one point, sharing its slope with that point.
Chain Rule: The chain rule allows us to find derivatives of composite functions by breaking them down into simpler functions and applying differentiation rules.
Implicit Differentiation: Implicit differentiation is used when we have equations where y cannot be easily expressed explicitly as a function of x.
"Derivative" appears in:
Study guides (10)
AP Calculus AB/BC - 4.3 Rates of Change in Applied Contexts other than Motion
AP Calculus AB/BC - 5.1 Using the Mean Value Theorem
AP Calculus AB/BC - 5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points
AP Calculus AB/BC - 5.3 Determining Intervals on Which a Function is Increasing or Decreasing
AP Calculus AB/BC - 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
AP Calculus AB/BC - 5.9 Connecting a Function, Its First Derivative, and its Second Derivative
AP Calculus AB/BC - 7.2 Verifying Solutions for Differential Equations
AP Calculus AB/BC - 7.5 Approximating Solutions Using Euler’s Method
AP Calculus AB/BC - 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
AP Calculus AB/BC - 9.1 Defining and Differentiating Parametric Equations
Additional resources (3)
Practice Questions (20+)
What is the derivative of a function at the point x = a equal to?
Which of the following best defines the derivative of a function?
What does the derivative of a function measure?
What notation is commonly used to represent the derivative of a function at a point x?
f(x) = x^3 + 2x. What is the derivative of the function at any point x?
f(x) = 5x^3 - 2x^2 + 4x - 1. What is the derivative of the function at any point x?
g(x) = e^x + ln(x). What is the derivative of the function at any point x?
f(x) = 4x^2 + 3x + 2. What is the derivative of the function at any point x?
g(x) = 3x^4 - 2x^3 + 5x^2 - 7x + 1. What is the derivative of the function at any point x?
f(x) = sqrt(x) + 2/x. What is the derivative of the function at any point x?
How can you graphically understand the derivative?
Why might the derivative of a function be positive at a specific point?
What is the estimate of the derivative of f(x) = x^2 at x = 2 using the difference quotient method?
What is the estimate of the derivative of g(x) = 3x^2 + 2x at x = 1 using the graph of the function?
What is the estimate of the derivative of h(x) = (x^3 + 2x)/(x+1) at x = 2 using technology?
Using technology, estimate the derivative of the function f(x) = 2x^3 - 4x^2 + 3x - 1 at the point x = 2. What is the numerical value of the estimated derivative?
If a function is smooth and has no "corners" or "breaks" in its graph, then its derivative will exist at:
What is the derivative of f(x) = √x?
What is the derivative of x^12?
What is the derivative of x^7?
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