Derivatives
Definition
Derivatives are the rates at which quantities change. They measure how a function behaves as its input (x-value) changes.
Analogy
Think of derivatives as the "speedometer" of a function. Just like a speedometer tells you how fast you're going at any given moment, derivatives tell you how fast a function is changing at any specific point.
Related terms
Tangent Line: A tangent line is a straight line that touches a curve at only one point and has the same slope as the curve at that point.
Rate of Change: Rate of change refers to how quickly one quantity changes in relation to another quantity.
Instantaneous Velocity: Instantaneous velocity is the derivative of position with respect to time. It represents an object's velocity at an exact moment in time.
"Derivatives" appears in:
Study guides (8)
AP Calculus AB/BC - 1.1 Introducing Calculus: Can Change Occur at An Instant?
AP Calculus AB/BC - 1.7 Selecting Procedures for Determining Limits
AP Calculus AB/BC - 2.2 Defining the Derivative of a Function and Using Derivative Notation
AP Calculus AB/BC - 2.7 Derivatives of cos x, sinx, e^x, and ln x
AP Calculus AB/BC - 2.10 Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions
AP Calculus AB/BC - 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration
AP Calculus AB/BC - 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled
AP Calculus AB/BC - 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions
Practice Questions (8)
- How can technology such as a calculator or software be used to estimate derivatives?
- Why is estimating derivatives an important concept in Calculus?
- What notation is commonly used for derivatives in implicit differentiation?
- How can we find the derivatives of inverse trigonometric functions?
- What kind of problems can derivatives be helpful for?
- What is the relationship between L'Hôpital's Rule and derivatives?
- How can relative extrema be identified by analyzing the derivatives of a function?
- The motion of a rolling ball on the coordinate plane is given by the set of parametric equations x(t) = 12sin(t) and y(t) = 6t^2. Which of the following derivatives is incorrect?
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