Differentiation finds the rate of change of a function — essentially, how steeply a curve is rising or falling at any given point. For a function like f(x) = xⁿ, the derivative (found using the power rule) is f'(x) = nxⁿ⁻¹, which tells you the gradient of the curve at any x-value. Unlike a straight line, which has one constant gradient everywhere, a curve's gradient changes continuously, so differentiation gives a formula for that changing gradient.
Example
If the height of a ball thrown in the air is modelled by h(t) = -5t² + 20t, differentiating gives h'(t) = -10t + 20, the ball's velocity at any time t — at t = 2, the velocity is h'(2) = -10(2) + 20 = 0, meaning the ball is momentarily at its highest point.
Key terms
Derivative:
A function describing the rate of change (gradient) of another function.
Power rule:
A rule for differentiating xⁿ: the derivative is nxⁿ⁻¹.
Gradient:
The steepness or slope of a curve at a given point.
Questions
1. Differentiation finds a function's:
Rate of change
Total area only
Starting value only
Exact colour on a graph
2. For f(x) = xⁿ, the power rule gives the derivative:
f'(x) = nxⁿ⁻¹
f'(x) = xⁿ⁺¹
f'(x) = n + x
f'(x) = x/n
3. Gradient describes:
The steepness of a curve at a point
The colour of a graph
The total number of points on a curve
A type of equation with no graph
4. Unlike a straight line, a curve's gradient:
Changes continuously along the curve
Is always exactly the same everywhere
Never exists at all
Is always exactly zero
5. If f(x) = x², the derivative f'(x) equals:
2x
x
2x²
x²
6. If f(x) = x³, the derivative f'(x) equals:
3x²
x²
3x
x³
7. The derivative of a function is itself:
A function describing the rate of change
Always just a single fixed number
Never usable for further calculation
Only relevant to straight lines
8. If f(x) = x⁴, the derivative f'(x) equals:
4x³
x³
4x
x⁴
9. If h(t) = -5t² + 20t, then h'(t) equals:
-10t + 20
-5t + 20
-10t
20t
10. Using h'(t) = -10t + 20, the velocity at t = 2 is:
0
10
20
-10
11. If f(x) = 5x² + 3x, then f'(x) equals:
10x + 3
5x + 3
10x
5x²
12. A derivative equal to zero at a specific point on a curve typically indicates:
A turning point, like a maximum or minimum
A mathematical error
The curve does not exist there
The steepest point on the entire curve
13. Why is finding where a derivative equals zero useful for a ball's height function like h(t) = -5t² + 20t?
It identifies the moment the ball's velocity is momentarily zero, which is its highest point
A derivative of zero always means the ball has landed
This calculation reveals nothing meaningful about the ball's motion
Zero derivatives only ever occur at the very start of motion
14. If f(x) = 4x³ - 2x, then f'(x) equals:
12x² - 2
4x² - 2
12x² - 2x
3x² - 2
15. Why does the power rule reduce the exponent by one when differentiating (e.g. x³ becomes 3x²)?
Differentiation measures instantaneous rate of change, and this pattern reflects how the rate of a power function behaves
The exponent is reduced purely as an arbitrary convention with no mathematical reason
Exponents are always meant to increase, not decrease, during differentiation
The power rule has no consistent, logical pattern behind it
16. If a curve's gradient is positive and decreasing as x increases, what does this suggest about the curve's shape?
The curve is still rising, but flattening out — possibly approaching a maximum
The curve is definitely falling steeply
The curve has no meaningful shape at this point
The gradient being positive means nothing about the curve's direction
17. Why is calculus (including differentiation) considered essential for modelling motion, like velocity and acceleration?
Velocity is the rate of change of position, and acceleration is the rate of change of velocity — both are derivatives
Motion can always be fully described without any reference to rates of change
Differentiation has no real connection to physical motion
Velocity and acceleration are unrelated to any mathematical rate of change
18. If h(t) = -5t² + 20t models a ball's height, what is the maximum height reached?
20
0
5
15
19. Why might businesses use differentiation to find a "marginal cost" or "marginal revenue" function?
It shows how cost or revenue changes with each additional unit, helping optimise production decisions
Marginal cost has no connection to rates of change or differentiation
Businesses never use calculus in any real decision-making
Differentiation can only ever apply to physical motion, not economics
20. If f(x) = 2x³ - 6x, at what value of x does f'(x) = 0 (other than x = -1)?
1
0
2
3
21. Why does the derivative of a straight line function (like f(x) = 3x + 2) always give a constant number, unlike the derivative of a curve?
A straight line has the same gradient at every point, so its rate of change never varies
Straight lines never have a defined gradient at all
Constant derivatives only ever occur for curves, not straight lines
The gradient of a straight line always changes at every single point
Answer key (parent copy)
1. Rate of change
2. f'(x) = nxⁿ⁻¹
3. The steepness of a curve at a point
4. Changes continuously along the curve
5. 2x
6. 3x²
7. A function describing the rate of change
8. 4x³
9. -10t + 20
10. 0
11. 10x + 3
12. A turning point, like a maximum or minimum
13. It identifies the moment the ball's velocity is momentarily zero, which is its highest point
14. 12x² - 2
15. Differentiation measures instantaneous rate of change, and this pattern reflects how the rate of a power function behaves
16. The curve is still rising, but flattening out — possibly approaching a maximum
17. Velocity is the rate of change of position, and acceleration is the rate of change of velocity — both are derivatives
18. 20
19. It shows how cost or revenue changes with each additional unit, helping optimise production decisions
20. 1
21. A straight line has the same gradient at every point, so its rate of change never varies