Glossary
Acceleration (in motion)
The rate of change of an object's velocity with respect to time, indicating how quickly its velocity is changing. It is the second derivative of the position function.
Example:
When a car brakes suddenly, it experiences negative Acceleration, causing its velocity to decrease.
Chain Rule
A rule for differentiating composite functions, stating that the derivative of f(g(x)) is f'(g(x)) * g'(x).
Example:
When finding the derivative of y = (3x - 5)⁴, the Chain Rule is essential because it's a function within a function.
Derivatives of Exponential Functions
The specific rules for finding the derivatives of exponential functions, such as e^x and a^x.
Example:
If you're modeling population growth with P(t) = 100e^(0.05t), finding P'(t) requires knowledge of Derivatives of Exponential Functions.
Derivatives of Inverse Functions
A rule stating that if g(x) is the inverse of f(x), then g'(x) = 1 / f'(g(x)).
Example:
If you know the derivative of f(x) = x³ + x, you can use the formula for Derivatives of Inverse Functions to find the derivative of its inverse at a specific point without explicitly finding the inverse function.
Derivatives of Inverse Trigonometric Functions
The specific rules for finding the derivatives of arcsin, arccos, arctan, arccot, arcsec, and arccsc functions.
Example:
When you encounter a problem involving arctan(2x), you'll need to recall the rules for Derivatives of Inverse Trigonometric Functions.
Derivatives of Logarithmic Functions
The specific rules for finding the derivatives of natural logarithms and logarithms with other bases.
Example:
The derivative of ln(x) is 1/x, a fundamental rule when working with Derivatives of Logarithmic Functions.
Derivatives of Trigonometric Functions
The specific rules for finding the derivatives of sine, cosine, tangent, cotangent, secant, and cosecant functions.
Example:
Knowing that the derivative of sin(x) is cos(x) is a key part of mastering Derivatives of Trigonometric Functions.
Implicit Differentiation
A technique used to find the derivative of a function that is not explicitly defined in terms of one variable, by differentiating both sides of an equation with respect to a variable and solving for the derivative.
Example:
For an equation like x² + y² = 9, you use Implicit Differentiation to find dy/dx, treating y as a function of x.
Instantaneous Rate of Change
The rate at which a quantity is changing at a specific moment in time, represented by the derivative of the function at that point.
Example:
If a rocket's altitude is given by h(t), its vertical speed at exactly t=10 seconds is its Instantaneous Rate of Change of altitude.
Position (in motion)
The location of a particle or object at a specific time, typically denoted by x(t) or s(t).
Example:
If a car is moving along a straight road, its distance from a starting point at any given moment is its Position.
Power Rule
A fundamental rule for differentiating functions of the form x^n, stating that the derivative is nx^(n-1).
Example:
To find the derivative of f(x) = x^7, you simply apply the Power Rule to get f'(x) = 7x^6.
Product Rule
A rule used to find the derivative of a function that is the product of two differentiable functions, given by (uv)' = u'v + uv'.
Example:
If you have a function like g(x) = x³ * cos(x), you'd use the Product Rule to find its derivative.
Quotient Rule
A rule used to find the derivative of a function that is the ratio of two differentiable functions, given by (u/v)' = (u'v - uv')/v².
Example:
To differentiate h(x) = (sin(x))/(x² + 1), the Quotient Rule is your go-to method.
Slope of the Tangent Line
The slope of the line that touches a curve at a single point, representing the instantaneous rate of change of the function at that point.
Example:
To find how steep the roller coaster track is at a specific point, you would calculate the Slope of the Tangent Line to its path at that point.
Speed (in motion)
The magnitude of an object's velocity, indicating how fast it is moving regardless of direction. It is always a non-negative value.
Example:
If a car's velocity is -60 mph, its Speed is 60 mph, indicating how fast it's going without regard to direction.
Velocity (in motion)
The rate of change of an object's position with respect to time, indicating both its speed and direction. It is the first derivative of the position function.
Example:
A positive Velocity means a particle is moving to the right, while a negative velocity means it's moving to the left.