Glossary
Algebraic Manipulation
The process of rearranging or simplifying algebraic expressions using fundamental algebraic properties and operations to achieve a desired form.
Example:
After applying differentiation rules, extensive algebraic manipulation is often needed to simplify the expression for into its most concise form.
Chain Rule
A rule used to differentiate composite functions. It states that the derivative of f(g(x)) is f'(g(x)) * g'(x).
Example:
To find the derivative of sin(x²), you apply the chain rule, differentiating the sine function first, then multiplying by the derivative of x².
Chain Rule
A fundamental rule in calculus for differentiating composite functions, stating that the derivative of \(f(g(x))\) is \(f'(g(x)) \cdot g'(x)\).
Example:
When differentiating , the chain rule is applied to handle the outer power function and the inner polynomial.
Critical Point
A point on the graph of a function where its derivative is either zero or undefined. These points are crucial for identifying local extrema and points of inflection.
Example:
When analyzing the trajectory of a projectile, the highest point it reaches is a critical point where its vertical velocity (derivative) momentarily becomes zero.
First Derivative
The derivative of a function, representing the instantaneous rate of change or the slope of the tangent line at any given point on the curve.
Example:
If a car's position is given by a function, its first derivative tells you the car's instantaneous velocity.
First Derivative
The result of differentiating a function once, representing the instantaneous rate of change or the slope of the tangent line at any point.
Example:
The first derivative of a cost function can tell a company the marginal cost of producing one more unit.
Horizontal Tangent
A tangent line to a curve that is perfectly flat, indicating that the rate of change of the dependent variable with respect to the independent variable is zero.
Example:
At the peak of a roller coaster's first hill, the track has a horizontal tangent, meaning the instantaneous vertical change is zero.
Implicit Differentiation
A technique used to find the derivative of an implicit function by differentiating both sides of the equation with respect to a variable, typically x, and applying the chain rule to terms involving y.
Example:
When working with equations like , implicit differentiation is essential to find .
Implicit Equation
An equation where the dependent variable (e.g., y) is not explicitly expressed as a function of the independent variable (e.g., x), but rather defined by a relationship between them.
Example:
The equation of a circle, x² + y² = r², is an implicit equation because y is not isolated on one side.
Implicit Function
A function where the dependent variable (y) is not explicitly isolated on one side of the equation, often defined by an equation involving both x and y.
Example:
The equation defines an implicit function that cannot be easily solved for y in terms of x.
Local Maximum
A point on a function's graph where the function's value is larger than at all nearby points. At such a point, the first derivative is zero and the second derivative is negative (or changes from positive to negative).
Example:
The highest point on a small hill within a larger mountain range is a local maximum of altitude.
Local Minimum
A point on a function's graph where the function's value is smaller than at all nearby points. At such a point, the first derivative is zero and the second derivative is positive (or changes from negative to positive).
Example:
The lowest point in a valley on a topographical map represents a local minimum in elevation.
Point of Inflection
A point on a curve where the concavity changes (from concave up to concave down, or vice versa). At this point, the second derivative is zero or undefined, and its sign must change.
Example:
On a graph showing the spread of a disease, the point of inflection often indicates when the rate of new infections begins to slow down after an initial rapid increase.
Product Rule
A rule for differentiating the product of two functions. If h(x) = f(x)g(x), then h'(x) = f'(x)g(x) + f(x)g'(x).
Example:
When finding the derivative of x * e^x, you use the product rule to account for both functions being multiplied.
Quotient Rule
A rule for differentiating a function that is the ratio of two other functions. If h(x) = f(x)/g(x), then h'(x) = (f'(x)g(x) - f(x)g'(x)) / (g(x))².
Example:
To differentiate a rational function like (x² + 1) / (x - 3), the quotient rule is essential.
Quotient Rule
A rule for differentiating a function that is the ratio of two other functions, given by the formula \( \left( \frac{u}{v} ight)' = \frac{u'v - uv'}{v^2} \).
Example:
To find the derivative of , you must use the quotient rule.
Second Derivative
The derivative of the first derivative, which provides information about the concavity of a function and helps classify critical points as local maxima, minima, or points of inflection.
Example:
For the car's motion, the second derivative of its position function gives its acceleration, indicating how its velocity is changing.
Second Derivative
The derivative of the first derivative of a function, representing the rate of change of the slope of the tangent line. It is often used to determine concavity or acceleration.
Example:
To find the acceleration of a particle whose position is given by , you would calculate the second derivative .
Vertical Tangent
A tangent line to a curve that is perfectly upright, indicating that the rate of change of the independent variable with respect to the dependent variable is zero, or the derivative dy/dx is undefined.
Example:
The curve x = y³ has a vertical tangent at the origin, where the slope becomes infinite.