Glossary
Chain Rule
A fundamental differentiation rule used for finding the derivative of composite functions. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x).
Example:
When differentiating sin(x³), the chain rule gives cos(x³) * (3x²).
Explicit Equations
Equations where one variable is isolated and directly expressed in terms of another variable, typically in the form y = f(x). These are straightforward to differentiate using standard rules.
Example:
y = 3x² - 5x + 7 is an explicit equation where y is clearly defined in terms of x.
Horizontal Tangent
A tangent line to a curve that has a slope of zero. This occurs at points where the derivative of the function is equal to zero, often indicating a local maximum or minimum.
Example:
The function f(x) = x³ - 3x has a horizontal tangent at x = 1 and x = -1, where f'(x) = 0.
Implicit Differentiation
A method used to find the derivative of an equation where one variable (often y) is not explicitly defined as a function of the other variable (often x). It involves differentiating both sides of the equation with respect to x and then solving for dy/dx.
Example:
To find the slope of the unit circle x² + y² = 1, we use implicit differentiation to get dy/dx = -x/y.
Point-Slope Form
A common algebraic form for the equation of a straight line, given by y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is the slope.
Example:
If a line passes through (1, 5) with a slope of -2, its equation in point-slope form is y - 5 = -2(x - 1).
Product Rule
A differentiation rule used to find the derivative of a product of two functions. If h(x) = u(x)v(x), then h'(x) = u'(x)v(x) + u(x)v'(x).
Example:
To differentiate x²cos(x)*, the product rule yields 2xcos(x) + x²(-sin(x)).
Tangent Line
A straight line that touches a curve at a single point and has the same slope as the curve at that specific point. Its slope is given by the derivative evaluated at that point.
Example:
The tangent line to the curve y = x² at the point (2, 4) has a slope of 4, so its equation is y - 4 = 4(x - 2).