Leibnitz's rule

Leibnitz's rule

1. Full Leibniz Rule — Variable Limits

For $F(α)= ∫_{a(α)}^{b(α)} f(x,α) dx$ where $f, f_α$ continuous and $a(α), b(α)$ differentiable:

i.e. partial differentiation inside + boundary contributions from moving limits. For constants $a',b'=0$ it reduces to previous topic.

Special case $F(x)= ∫_{a(x)}^{b(x)} f(t) dt$ (integrand independent of $x$):

is the second fundamental theorem combined with chain rule.

2. Worked Examples

(a) F(α)= ∫_{0}^{α} sin(α x)/x dx? Need handle. Simpler:
    F(α)= ∫_{0}^{α²} e^{α x} dx = [e^{α x}/α]_{0}^{α²}= (e^{α³}-1)/α
    Via Leibniz: f=e^{α x}, f_α= x e^{α x}, b=α² ⇒ dF/dα= ∫_{0}^{α²} x e^{α x}dx + e^{α·α²}·2α - e^{0}·0
    Differentiate direct (1+α³)... matches ✓

(b) F(x)= ∫_{x}^{x²} t³ dt = x^8/4 - x^4/4
    Leibniz: f=t³, a=x, b=x² ⇒ F'= (x²)³·2x - x³·1 =2x^7 - x^3 matches differentiate ✓

(c) Evaluate I(α)= ∫_{0}^{π} log(1+α cos x)/cos x?? Use Leibniz to get ODE for I(α)

(d) F(x)= ∫_{0}^{x} (x - t) f(t) dt ⇒ F' = ∫_{0}^{x} f(t) dt, F''=f(x) (differentiate twice) ✓

(e) Classic: d/dx ∫_{0}^{x} sin(t²) dt = sin(x²) ; d/dx ∫_{0}^{x²} sin(t²) dt = sin(x⁴)·2x

3. Repeated Differentiation

Apply again to get higher derivatives: e.g., $d²F/dα² = ∫ f_{αα} dx + 2 f(b)??... plus terms from limits. Remember to differentiate boundary terms as well.

4. Exam Points

• State full Leibniz $d/dα ∫_{a(α)}^{b(α)} f(x,α)dx$ — 5 marks, very frequent.
• Distinguish from fixed-limit case — 3 marks.
• Evaluate $d/dx ∫_{x}^{x²} e^{t²} dt$, $d/dx ∫_{0}^{x} (x-t)sin t dt$ type — 5-7 marks, very frequent.
• Use Leibniz to prove $d/dx ∫_{0}^{x} f(t)(x-t)^{n-1}/(n-1)! dt = ∫_{0}^{x} f(t)(x-t)^{n-2}/(n-2)! dt$ — 5 marks, frequent.