Derivatives¶
Recall (doc/postdoc). Not a first course.
Job: best linear approximation; Fréchet derivative \(f'(a)\in L(\mathbb{R},\mathbb{R})\cong\mathbb{R}\).
Core¶
\[
f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}
\]
| Rule | Formula |
|---|---|
| Linearity | \((af+bg)'=af'+bg'\) |
| Product | \((uv)'=u'v+uv'\) |
| Quotient | \((u/v)'=(u'v-uv')/v^2\) |
| Chain | \((f\circ g)'=(f'\circ g)\,g'\) |
| Inverse | \((f^{-1})'(y)=1/f'(x)\) if \(f'(x)\neq0\) |
| Power / exp / ln | \((x^n)'=nx^{n-1}\), \((e^x)'=e^x\), \((\ln x)'=1/x\) |
MVT: \(f\) cont. \([a,b]\), diffble \((a,b)\) \(\Rightarrow\) \(\exists c\) with \(f'(c)=\frac{f(b)-f(a)}{b-a}\).
Taylor: \(f(x)=\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a)^k+R_n\); Lagrange \(R_n=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}\).
Used later for¶
- Integrals: FTC
- ODE: \(y'=F(x,y)\)
- Optimization: critical points, Hessian
- Differential geometry: tangent maps
- FA: Gateaux/Fréchet
Check¶
Is \(f(x)=x^2\sin(1/x)\) (\(f(0)=0\)) differentiable at \(0\) with continuous \(f'\)?
Differentiable everywhere with \(f'(0)=0\); \(f'\) discontinuous at \(0\) (oscillation). Differentiability \(\not\Rightarrow\) \(C^1\).