A curve's gradient is the gradient of its tangent: A straight line has one gradient everywhere.
A curve is different — its steepness changes.
The gradient at a point is defined as the gradient of the tangent (the line that just touches the curve there).
The derivative gives that gradient.
The tangent to y = x² at a point — its gradient is the derivative there. As the point moves, the tangent's gradient changes.
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IB-style question — the gradient changes
On the curve y = x², the gradient at (1, 1) is 2 and at (3, 9) it is 6.
Explain why a curve does not have a single gradient.
Step by step
- Compare the two gradients.
- So the steepness depends on where you are.
Final answer
Because the gradients differ (2 vs 6), the curve gets steeper as x increases — there is no single gradient, only a gradient at each point.
Tangent, not chord: The gradient at a point uses the tangent there — not a chord joining two separate points.
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How fast y changes as x changes: The derivative is also a rate of change: how fast y changes per unit of x, at an instant.
If y depends on time, the derivative is the instantaneous rate (e.g. speed = rate of change of distance).
IB-style question — interpret a rate
The volume V (litres) of water in a tank after t minutes has dV/dt = 12 at t = 5.
Interpret this value.
Step by step
- dV/dt is the rate V changes per minute.
- At t = 5 specifically.
Final answer
At t = 5 minutes, the water volume is increasing at 12 litres per minute.
Carry the units: A rate of change has units of (y-units) per (x-unit) — e.g. litres per minute, metres per second.
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f'(x) and dy/dx mean the gradient function: The derivative is written f'(x) ("f prime of x") or dy/dx.
It is itself a function — feed in an x and it returns the gradient there.
(How to find it is the next topic; here we just use it.)
IB-style question — gradient at a point
The gradient function of y = x² is f'(x) = 2x.
Find the gradient of the curve at x = 3.
Step by step
- Substitute x = 3 into the gradient function.
- Evaluate.
Final answer
The gradient at x = 3 is 6.
Gradient at a point = substitute into f'(x): To get a number, substitute the x-value into the gradient function f'(x).
Positive up, negative down, zero flat: The sign of f'(x) tells you the shape: f'(x) > 0 → the curve is increasing (going up); f'(x) < 0 → decreasing (going down); f'(x) = 0 → a stationary point (momentarily flat).
IB-style question — read the sign
For a function f, f'(2) = 5 and f'(−1) = −3.
State whether f is increasing or decreasing at each point.
Step by step
- At x = 2 the derivative is positive.
- At x = −1 the derivative is negative.
Final answer
Increasing at x = 2; decreasing at x = −1.
f'(x) = 0 → flat: A zero gradient means a stationary point — a peak, trough or a flat spot (explored in 5.8).