Definition of derivative
Definition of a Derivative
Definition of a Derivative is the opening topic of Differentiation in the California Calculus Standards. It is aligned to Standard 4.0, which requires students to understand the formal definition of the derivative at a point, its meaning as the slope of the tangent line (Standard 4.1), and as an instantaneous rate of change (Standard 4.2).
The derivative \(f'(a)=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}\) measures how fast a function changes at a point. It is the slope of the tangent line and the instantaneous rate of change, and it underlies the rest of differential calculus.
Theory
The derivative \(f'(a)\) measures how fast a function is changing at \(x=a\). It is defined as a limit of slopes, \(f'(a)=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}\), and it equals both the slope of the tangent line and the instantaneous rate of change at that point.
The derivative of \(f\) at \(x=a\) is the limit of the difference quotient — the slope of the secant line as the two points slide together:
Geometrically, each difference quotient is the slope of a secant line; as \(h\to 0\) those slopes approach the slope of the tangent line. That tangent slope is \(f'(a)\).
The same number is the instantaneous rate of change of \(f\) at \(a\) — for a position function, that rate is velocity.
The limit definition (at a point, and as a function):
The tangent line at \(\big(a,f(a)\big)\) uses the point-slope form:
How to differentiate from the definition
- Form the difference quotient \(\dfrac{f(x+h)-f(x)}{h}\).
- Expand and simplify until every term has a factor of \(h\) to cancel with the bottom.
- Take the limit \(h\to 0\) in the simplified expression.
Expand \((x+h)^2\) and simplify before the \(h\to 0\) step.
| \(\dfrac{(x+h)^2-x^2}{h}\) | \(=\) | \(\dfrac{x^2+2xh+h^2-x^2}{h}\) |
| \(=\) | \(\dfrac{2xh+h^2}{h}=2x+h\) | |
| \(\lim_{h\to 0}(2x+h)\) | \(=\) | \(2x\) |
The slope is \(f'(3)\), and the point is \((3,9)\).
| \(f'(x)\) | \(=\) | \(2x\) |
| \(f'(3)\) | \(=\) | \(6\) |
| \(y-9\) | \(=\) | \(6(x-3)\) |
So the tangent line is \(y=6x-9\).
Velocity is the derivative of position, \(s'(t)\).
| \(s'(t)\) | \(=\) | \(2t\) |
| \(s'(4)\) | \(=\) | \(8\) |
The velocity is \(8\) ft/s.
Combine the two fractions over a common denominator first.
| \(\dfrac{\dfrac{1}{x+h}-\dfrac{1}{x}}{h}\) | \(=\) | \(\dfrac{x-(x+h)}{h\,x(x+h)}\) |
| \(=\) | \(\dfrac{-1}{x(x+h)}\) | |
| \(\lim_{h\to 0}\) | \(=\) | \(-\dfrac{1}{x^2}\) |
Common pitfalls
Frequently asked questions
What is the definition of a derivative?
The limit of the difference quotient: \(f'(a)=\lim_{h\to 0}\dfrac{f(a+h)-f(a)}{h}\). It is the slope of the tangent line at \(a\).
What does the derivative represent?
The instantaneous rate of change of the function, which equals the slope of the tangent line at that point. For position, it is velocity.
How do you find the equation of a tangent line?
Compute the slope \(f'(a)\), find the point \((a,f(a))\), then use point-slope form \(y-f(a)=f'(a)(x-a)\).
When is a function not differentiable?
Where the difference-quotient limit fails: at a sharp corner, a cusp, a jump, or a vertical tangent. A function must at least be continuous to be differentiable.