Introduction to Differentiation
Differentiation finds the instantaneous rate of change of a function, which corresponds to the gradient of the tangent line at any point.
What You Need to Know
Key Concept Diagram
The derivative f'(x) represents the gradient of the tangent to the curve y = f(x) at any point
Power rule: d/dx(xⁿ) = nxⁿ⁻¹ — multiply by the exponent then reduce it by one
The derivative of a constant is zero; the derivative of a linear term ax is simply a
Higher-order derivatives measure rates of change of the gradient itself
Key Vocabulary
Derivative
The function that gives the instantaneous rate of change of another function
Gradient
The slope of the tangent line to a curve at a specific point
Tangent
A straight line that touches a curve at exactly one point and has the same slope as the curve there
Stationary Point
A point where the derivative equals zero, indicating a local maximum, minimum, or inflection
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the derivative of f(x) = 5x⁴?
Question 2
At a stationary point, the value of the derivative is:
Question 3
Find the gradient of y = 3x² − 2x at x = 2.
Key Concepts Summary
- ●The derivative f'(x) represents the gradient of the tangent to the curve y = f(x) at any point
- ●Power rule: d/dx(xⁿ) = nxⁿ⁻¹ — multiply by the exponent then reduce it by one
- ●The derivative of a constant is zero; the derivative of a linear term ax is simply a
- ●Higher-order derivatives measure rates of change of the gradient itself