01 Key Concepts
The Derivative as a Slope
f'(x) gives the slope of the tangent line to the graph of f at any point x.
Definition via Limits
f'(x) = lim as h approaches 0 of [f(x+h) - f(x)] / h.
Notation
Common notations include f'(x), dy/dx, and d/dx[f(x)] — they all mean the same thing.
Differentiability
A function must be continuous to be differentiable, but continuity alone does not guarantee differentiability (sharp corners, for example).
Power Rule Preview
For f(x) = x^n, f'(x) = n*x^(n-1). This is the single most-used shortcut in differentiation.
02 Key Formulas
- f'(x) = lim (h->0) [f(x+h) - f(x)] / h
- d/dx [x^n] = n*x^(n-1)
03 Solved Examples
Example 1 Use the limit definition to find the derivative of f(x) = x^2.
- f(x+h) = (x+h)^2 = x^2 + 2xh + h^2.
- f(x+h) - f(x) = 2xh + h^2.
- Divide by h: (2xh+h^2)/h = 2x + h.
- Take the limit as h approaches 0: 2x + 0.
Answer: f'(x) = 2x
Example 2 Find the derivative of f(x) = x^3 using the power rule.
- Apply d/dx[x^n] = n*x^(n-1) with n=3.
- f'(x) = 3*x^(3-1) = 3x^2.
Answer: f'(x) = 3x^2
Example 3 Find the slope of the tangent line to f(x) = x^2 at x = 4.
- From the earlier example, f'(x) = 2x.
- Substitute x = 4: 2(4).
Answer: Slope = 8
04 Practice Questions
1Find the derivative of f(x) = x^4.
4x^3
2Find the derivative of f(x) = x (i.e., x^1).
1
3Find the derivative of a constant function f(x) = 7.
0
4Find f'(2) if f(x) = x^2.
4
5What does f'(x) represent graphically?
The slope of the tangent line at x
📄 Differentiation — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.