Continuity and Differentiability
Differentiability, chain rule and implicit functions
Use the chain rule, implicit differentiation and inverse trig derivatives.
f is differentiable at c if its graph has a clear slope (tangent) there. Check it like continuity: the left-hand derivative and right-hand derivative must be finite and equal.
Differentiable ⟹ continuous, but not the other way round. |x| is continuous at 0, but its graph has a sharp corner there: slope –1 on the left and +1 on the right. So |x| is not differentiable at 0. Sharp corners and jumps kill differentiability.
Chain rule: for a function inside a function, differentiate the outside (leaving the inside alone), then multiply by the derivative of the inside.
- d/dx (2x + 3)⁵ = 5(2x + 3)⁴ × 2 = 10(2x + 3)⁴.
- d/dx sin(x²) = cos(x²) × 2x.
Implicit differentiation: when x and y are mixed and y cannot easily be written alone, differentiate both sides with respect to x. Every y term gets an extra dy/dx (chain rule). Then collect the dy/dx terms and solve.
Example: x² + y² = 25 gives 2x + 2y dy/dx = 0, so dy/dx = –x/y.
Inverse trig derivatives:
| Function | Derivative |
|---|---|
| sin⁻¹x | 1/√(1 – x²) |
| cos⁻¹x | –1/√(1 – x²) |
| tan⁻¹x | 1/(1 + x²) |
| cot⁻¹x | –1/(1 + x²) |
With the chain rule: d/dx sin⁻¹(3x) = 3/√(1 – 9x²). For complicated inverse trig expressions, simplify first with a substitution such as x = tan θ (Chapter 2).
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