Level 5 · Mathematics M5.4

Differentiation

Find how fast things change, and use it to find the highest and lowest points of curves.

Planned. This book is not written yet. Below is its plan, which a teacher can already use.

You are ready for this book if

You can find the gradient of the line through (1, 3) and (4, 9).

The plan

  1. M5.4.1 The gradient of a curve
    • Estimate a gradient using chords that get shorter and shorter
    • Find the gradient of y = x² from first principles
  2. M5.4.2 Differentiating powers
    • Differentiate xⁿ for any power n
    • Differentiate sums like 3x⁴ − 2x + 7
    • Rewrite √x and 1/x² as powers before you differentiate
  3. M5.4.3 Tangents and normals
    • Find the equation of the tangent at a point on a curve
    • Find the equation of the normal at a point on a curve
  4. M5.4.4 Turning points
    • Find turning points where dy/dx = 0
    • Use the second derivative to tell a maximum from a minimum
  5. M5.4.5 Biggest and smallest
    • Find the largest open box you can fold from a square sheet
    • Find the shortest fence for a pen of fixed area
  6. M5.4.6 Exponentials, logs and trig functions
    • Differentiate eˣ, ln x, sin x and cos x
    • Explain why x must be in radians for these results
  7. M5.4.7 Chain, product and quotient rules
    • Differentiate (3x + 1)⁵ with the chain rule
    • Differentiate x² sin x with the product rule
    • Differentiate a fraction with the quotient rule
  8. M5.4.8 Connected rates and implicit curves
    • Find how fast a circle's area grows as its radius grows
    • Differentiate an implicit equation like x² + y² = 25

For parents and teachers

Level 5 fits learners who can already: read a textbook. use functions and graphs. In many schools that is about 17 to 20, but start wherever the learner is.

Strand: Algebra. Written and drawn by attnom. Free to read, print and share.