NCEA Level 2The derivativeAS 91262

The derivative: what the gradient function means

What this lesson covers

  • The derivative gives the gradient of the curve at every point.
  • The gradient at a point is the gradient of the tangent there, which the chord gradients approach.
Key idea

How steep is a curve?

  • A straight line has one gradient everywhere.
  • A curve gets steeper or flatter as you go.
  • The derivative gives its gradient at every point.
Worked example

The gradient of y = x² at x = 3

  • Mark the point (3, 9) on the curve.
  • A chord joins it to a second point.
  • Chord to x = 4: gradient 7.
The gradient of y = x² at x = 3
Worked example

Move the second point closer

  • To x = 3.5: gradient 6.5.
  • To x = 3.1: gradient 6.1.
  • The chords are closing in on one line.
Move the second point closer
Worked example

The limit is the tangent

  • As the gap shrinks, the gradient heads to 6.
  • That line just touches the curve: the tangent.
  • Gradient at x = 3 is 6, which is 2 × 3.
The limit is the tangent
Worked example

The numbers behind the drawing

  • Rise ÷ run, each chord.
  • The run shrinks to 0.
  • Gradients close in on 6.
  1. gradient = rise ÷ run
  2. x = 4: (16 − 9) ÷ 1 = 7
  3. x = 3.5: (12.25 − 9) ÷ 0.5 = 6.5
  4. x = 3.1: (9.61 − 9) ÷ 0.1 = 6.1
  5. x = 3.01: (9.0601 − 9) ÷ 0.01
  6. = 0.0601 ÷ 0.01 = 6.01
  7. gradients → 6 as run → 0
Key idea

A gradient for every x

  • Repeat at any point on y = x²: the gradient is 2x.
  • dy/dx = 2x is the gradient function.
  • It is a new function, not the curve.
y = x²the curve: heights
dy/dx = 2xthe gradient at every x
at x = 3height 9, gradient 6
at x = −1height 1, gradient −2
Key idea

The rule for xⁿ

  • Bring the power down in front.
  • Then reduce the power by one.
  • A constant term has gradient 0.
ydy/dx
x²2x
x³3x²
x⁵5x⁴
5x5
7 (a constant)0
Worked example

Using it at a point

  • Differentiate first.
  • Brackets around −2.
  • Positive: the curve rises.
  1. y = x³
  2. dy/dx = 3x²power down, one less
  3. x = −2: 3 × (−2)²into dy/dx
  4. = 3 × 4 = 12
  5. x = 1: 3 × 1² = 3
Worked example

The most common gradient mistake

  • Substituting into y gives a height, not a gradient.
  • For a gradient, substitute into dy/dx.
  • Write which one you are using.
Askedthe gradient of y = x³ at x = 2
Wrongsubstitute into y: 8
Rightdy/dx = 3x², so 3 × 4 = 12
Key idea

A gradient is a rate of change

  • Units of y per unit of x.
  • Metres per second, dollars per item.
  • Say the rate in context, with units.
Summary

What to take away

  • The derivative is the gradient, everywhere at once.
Derivativethe gradient at every point
Tangentthe limit of the chords
Rulexⁿ becomes nxⁿ⁻¹
Constantdifferentiates to 0
Gradient at a pointsubstitute into dy/dx
Height at a pointsubstitute into y
Your turn

Your turn: four questions

  • Pause here and answer each one.
  • Question 3 is a chord, not a derivative.
  • Question 4: set the gradient equal to 10.
Question
1Differentiate y = x⁴.
2Find the gradient of y = x³ at x = −2.
3Gradient of the chord (1, 1) to (1.1, 1.21) on y = x²?
4Where does y = x² have a gradient of 10?
Show answer

1. dy/dx = 4x³. 2. dy/dx = 3x², so 3 × (−2)² = 12. 3. (1.21 − 1) ÷ (1.1 − 1) = 0.21 ÷ 0.1 = 2.1, close to the tangent gradient 2. 4. 2x = 10, so x = 5, at the point (5, 25).

Worked example

The answers

  1. 1. y = x⁴, dy/dx = 4x³power down, one less
  2. 2. dy/dx = 3x²
  3. x = −2: 3 × (−2)² = 3 × 4 = 12
  4. 3. (1.21 − 1) ÷ (1.1 − 1)rise ÷ run
  5. = 0.21 ÷ 0.1 = 2.1
  6. close to dy/dx = 2(1) = 2the tangent
  7. 4. dy/dx = 2x = 10
  8. x = 5, y = 5² = 25
  9. the point (5, 25)

Is your child on track?

Take the free placement test