Glossary
Concave Down
A portion of a function's graph where the curve opens downwards, resembling an inverted cup. This occurs when the second derivative is negative.
Example:
The function y = -x² is concave down everywhere because its second derivative, y'' = -2, is always negative.
Concave Up
A portion of a function's graph where the curve opens upwards, resembling a cup. This occurs when the second derivative is positive.
Example:
The parabola y = x² is concave up everywhere because its second derivative, y'' = 2, is always positive.
Constant Multiple Rule
This rule allows you to pull a constant factor out of the derivative operation; the derivative of a constant times a function is the constant times the derivative of the function.
Example:
To find the derivative of g(x) = 5sin(x), you use the Constant Multiple Rule to get g'(x) = 5cos(x).
Constant Rule
This rule states that the derivative of any constant value is always zero.
Example:
If f(x) = 7, then by the Constant Rule, f'(x) = 0, because a constant function has no change.
Derivative
The derivative of a function represents its instantaneous rate of change at a specific point. Geometrically, it is the slope of the tangent line to the function's graph at that point.
Example:
If a car's position is given by s(t) = t², its derivative s'(t) = 2t gives its instantaneous velocity at any time t.
Derivative of cos(x)
The derivative of the cosine function, cos(x), is -sin(x).
Example:
If the temperature oscillates as T(t) = cos(t), the rate of change of temperature is the derivative of cos(t), which is T'(t) = -sin(t).
Derivative of eˣ
The derivative of the natural exponential function, eˣ, is itself, eˣ.
Example:
If a population grows exponentially according to P(t) = 100eᵗ, its growth rate is the derivative of eᵗ, which is P'(t) = 100eᵗ.
Derivative of ln(x)
The derivative of the natural logarithm function, ln(x), is 1/x.
Example:
To find the rate of change of a logarithmic scale, such as sound intensity, you would use the derivative of ln(x), which is 1/x.
Derivative of sin(x)
The derivative of the sine function, sin(x), is cos(x).
Example:
If a particle's position is given by s(t) = sin(t), its velocity is the derivative of sin(t), which is v(t) = cos(t).
Derivative of tan(x)
The derivative of the tangent function, tan(x), is sec²(x).
Example:
When analyzing the slope of a tangent line for y = tan(x), you'd use the derivative of tan(x) to find y' = sec²(x).
Horizontal Tangent
A tangent line that is perfectly flat, indicating that the slope of the function at that point is zero. This often occurs at local maxima or minima.
Example:
To find where the function f(x) = x² - 4x has a horizontal tangent, you set its derivative f'(x) = 2x - 4 equal to zero and solve for x.
Instantaneous Rate of Change
This describes how quickly a quantity is changing at a precise moment in time, as opposed to its average rate of change over an interval.
Example:
Finding the instantaneous rate of change of a balloon's volume with respect to its radius when the radius is exactly 5 cm requires a derivative.
Power Rule
A fundamental rule for differentiating functions of the form xⁿ, where the derivative is nxⁿ⁻¹.
Example:
Using the Power Rule, the derivative of f(x) = x⁵ is f'(x) = 5x⁴.
Product Rule
Used to find the derivative of a product of two functions: (first function * derivative of second) + (second function * derivative of first).
Example:
To differentiate f(x) = x * cos(x), apply the Product Rule: f'(x) = (1 * cos(x)) + (x * -sin(x)) = cos(x) - xsin(x).
Quotient Rule
Used to find the derivative of a function that is a ratio of two other functions: (low d high - high d low) / (low squared).
Example:
To find the derivative of y = sin(x)/x, use the Quotient Rule: y' = (x * cos(x) - sin(x) * 1) / x².
Second Derivative
The derivative of the first derivative of a function, often denoted as f''(x) or d²y/dx². It describes the rate of change of the slope, indicating concavity.
Example:
If a car's position is s(t), its velocity is s'(t), and its acceleration is the second derivative, s''(t).
Sharp Corner/Cusp
A point on a graph where the function changes direction abruptly, preventing a unique tangent line from being defined, thus the derivative does not exist.
Example:
The absolute value function f(x) = |x| has a sharp corner (or cusp) at x=0, so its derivative does not exist there.
Slope of a Tangent Line
The slope of the tangent line at a point on a curve is numerically equal to the derivative of the function at that specific point.
Example:
To find the slope of a tangent line to the curve y = x³ at x=2, you'd calculate the derivative y' = 3x² and then evaluate it at x=2, getting 12.
Sum Rule
The derivative of a sum of functions is the sum of their individual derivatives.
Example:
If h(x) = x² + eˣ, then by the Sum Rule, h'(x) = 2x + eˣ.
Tangent Line
A straight line that touches a curve at a single point, sharing the same slope as the curve at that point.
Example:
The line y = 2x - 1 is the tangent line to the parabola y = x² at the point (1,1).
Vertical Tangent
A tangent line that is perfectly vertical, indicating an undefined or infinite slope at that point on the curve, meaning the derivative does not exist.
Example:
The function f(x) = x^(1/3) has a vertical tangent at x=0, where its derivative is undefined.