Continuity & differentiability
Standard derivatives, the chain and product rules, and continuity.
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To find how fast a combined function changes, use simple rules for each piece.
Look: y = |x| has no break, but the sharp corner at 0 has no single steepness.
✏️ Your turn!
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1d/dx (sin 3x)
2d/dx (x²)
3d/dx (x eˣ)
! Careful!
✗Forgetting to multiply by the inside’s derivative.
✓Chain rule: × 6x.
✗(uv)′ = u′ × v′.
✓Product rule: u′v + uv′.
See the whole lesson on one page
To find how fast a combined function changes, use simple rules for each piece.
Chain rule: derivative of the outside × derivative of the inside. A graph with no breaks is continuous; a sharp corner has no single steepness.
Think of it like thisA function inside a function is like nesting dolls. The chain rule: open the outside doll, then the inside one, and multiply.
Solved together
Find the derivative of (3x² + 1)⁵.
Outside: ( )⁵. Its derivative: 5( )⁴.
5(3x² + 1)⁴Inside: 3x² + 1. Its derivative: 6x.
6xMultiply them.
5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴
Answer30x(3x² + 1)⁴.
For older learners and grown-ups: a little more
Quotient rule: (u/v)′ = (u′v − uv′)/v². Example: d/dx [x/(x + 1)] = 1/(x + 1)².
Now practise it
Kanakku makes new questions on continuity & differentiability every time. It shows the working and gives a hint when you are stuck.