Glossary
Chain Rule
A fundamental differentiation rule used to find the derivative of composite functions, stating that d/dx [f(g(x))] = f'(g(x)) * g'(x).
Example:
When finding the derivative of h(x) = (g(x))², you must apply the chain rule to get h'(x) = 2g(x)g'(x).
Differentiable Function
A function is differentiable at a point if its derivative exists at that point, meaning it has a well-defined tangent line and no sharp corners or breaks.
Example:
The function f(x) = x³ is a differentiable function for all real numbers, allowing us to find its derivative at any point.
Inverse Function
An inverse function, denoted f⁻¹(x), 'undoes' the action of the original function f(x); if f(a) = b, then f⁻¹(b) = a.
Example:
For f(x) = x + 3, its inverse function is f⁻¹(x) = x - 3, as it reverses the addition.
Inverse Function Derivative Formula
This formula states that the derivative of an inverse function, f⁻¹(x), is the reciprocal of the derivative of the original function, f'(x), evaluated at the inverse function's output: (f⁻¹)'(x) = 1 / f'(f⁻¹(x)).
Example:
If you need to find the derivative of f⁻¹(x) at x=5, you'd first find f⁻¹(5), say it's a, then calculate 1/f'(a).
Invertible Function
A function is invertible if it is one-to-one, meaning each output value corresponds to exactly one input value, allowing for the existence of an inverse function.
Example:
The function f(x) = e^x is an invertible function because it passes the horizontal line test, ensuring a unique inverse.
Point-Slope Form
A standard form for the equation of a straight line, y - y₁ = m(x - x₁), where (x₁, y₁) is a point on the line and m is its slope.
Example:
To write the equation of a tangent line with slope 1/2 passing through (3, 2), you would use the point-slope form: y - 2 = (1/2)(x - 3).
Reciprocal of the Derivative
A memory aid emphasizing that the derivative of an inverse function is found by taking 1 divided by the derivative of the original function, evaluated at the appropriate point.
Example:
The phrase 'the derivative of the inverse is the reciprocal of the derivative' helps remember that if f'(x) = 2x, then the inverse's derivative will involve 1/(2*f⁻¹(x)).
Reflection over y = x
The geometric property that the graph of an inverse function is a mirror image of the original function's graph across the line y = x.
Example:
Graphing y = x² (for x ≥ 0) and its inverse y = √x clearly shows them as reflections over y = x.
Tangent Line
A straight line that touches a curve at a single point and has the same slope as the curve at that point, representing the instantaneous rate of change.
Example:
Finding the equation of the tangent line to g⁻¹(x) at x=2 requires knowing both the point (2, g⁻¹(2)) and the slope (g⁻¹)'(2).