MathematicsExam code: 0580

Differentiation

Coordinate Geometry and Graphs

PractiseFlashcards

What differentiating produces

  • A straight line has one gradient, but a curve is steeper in some places than others, so its gradient depends on where you are
  • Differentiating turns the equation of a curve into a second equation that gives the gradient at any point
  • That second equation is the gradient function, also called the derivative, and it is written dy/dx
  • Substituting an x value into dy/dx returns the gradient of the curve there, and the y value is never needed
  • Drawing a tangent and measuring it estimates the same number; differentiating gives it exactly
The curve y = x³ − 2x² + 6 with a straight tangent drawn touching it at a marked point P, showing that the gradient of the curve at P is the gradient of that tangent
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  • Write "y =" in front of the curve and "dy/dx =" in front of the derivative every time, so the two never get confused

The rule

  • Differentiating a power of x means multiplying by that power, then knocking one off it
    • x⁶ differentiates to 6x⁵
    • 3x⁴ differentiates to 12x³
  • In general, kxⁿ differentiates to knxⁿ⁻¹
The rule for differentiating a power set out in two steps: from y = kxⁿ, multiply by the power and then take one off the power, giving dy/dx = knxⁿ⁻¹
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  • Two cases look different but follow the same rule
    • A term in x alone loses its x: 7x differentiates to 7, which matches the gradient of the line y = 7x
    • A number on its own differentiates to 0, which matches the gradient of a horizontal line
y = 3x² − 5x + 3 differentiated term by term to dy/dx = 6x − 5, with an arrow noting that the constant 3 differentiates to zero and so leaves nothing behind
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  • Differentiate a sum term by term, taking each one separately and keeping the signs between them
  • This specification only asks for powers of x that are positive whole numbers or zero, and for sums of at most three such terms
The curve y = 2x³ + 3x² − 12x + 1 sketched on axes, crossing the x-axis three times with a peak to the left of the y-axis and a trough to the right
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Worked example

Differentiating and using the result

A curve has equation y = 4x³ + 7x − 2. Find dy/dx, and hence the gradient of the curve where x = 2.

Solution:

  • Differentiating turns the equation of a curve into its gradient function, which gives the gradient at any point rather than one fixed number
  • The rule is to multiply by the power and then reduce the power by one, applied to each term separately
  • 4x³: multiply by 3 and drop the power to 2, giving 12x²
  • 7x is 7x¹: multiply by 1 and drop the power to 0, and x⁰ = 1, so it gives 7
  • −2 is a constant with no x at all, so it differentiates to 0: a constant shifts the curve up or down without changing its steepness
  • So dy/dx = 12x² + 7
  • The gradient at a point comes from substituting that x into the derivative, not into the curve
  • At x = 2: 12 × 2² + 7 = 48 + 7 = 55

Worked example

Finding unknown powers and coefficients

A curve has equation y = axⁿ + 4x³ − 9, and its gradient function is dy/dx = 35x⁶ + bx². Find the values of a, n and b.

Solution:

  • Differentiate the curve term by term, keeping the letters: axⁿ gives naxⁿ⁻¹, 4x³ gives 12x², and −9 gives 0
  • So dy/dx = naxⁿ⁻¹ + 12x², which must match 35x⁶ + bx² term by term
  • Find the power first, because the number in front is a × n and cannot be split until n is known
  • Powers: n − 1 = 6, so n = 7
  • Numbers in front: na = 35, so 7a = 35 and a = 5
  • The x² terms: b = 12
  • a = 5, n = 7 and b = 12
  • Check: y = 5x⁷ + 4x³ − 9 differentiates to 35x⁶ + 12x², as given

Working backwards from a gradient

  • A question may give a gradient and ask where the curve has it
  • Set dy/dx equal to that number and solve the equation, which is usually a quadratic
  • Each solution is an x value, so put it back into the curve's own equation to obtain the matching y
  • Take care to read which side the number sits on: making the derivative equal 5 is a different instruction from replacing x by 5

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