Let A(t) be a smooth function of t taking values in Rⁿ ˣ ⁿ , and assume that A(s)A(t)=A(t)A(s) for all s and t. Show that d/dt eᴬ⁽ᵗ⁾ = A′ (t)eᴬ⁽ᵗ⁾ =eᴬ⁽ᵗ⁾ A′t). Hint. Show that if (1) holds, then d/dt eᴬ⁽ᵗ⁾ = d/ds (e ᴬ⁽ᵗ⁺ˢ⁾−ᴬ⁽ᵗ⁾eᴬ⁽ᵗ⁾)ₛ₌₀. Next, notice that, as a function of s, the matrix A(t+s)−A(t) satisfies the hypotheses of the previous problem. Use this and (3) to conclude the identity