Please explain how to solve step by step and name any rules or theorems used.

The following is from Chiang's fundamental mathematical economics.

1. Find the instantaneous rate of growth: (@) y = 512 (c) y = ob (e) y = 1/3' (b) y = at (d) y = 2'(12) 2. If population grows according to the function H = Ho(2)" and consumption by the function C = Coed, find the rates of growth of population, of consumption, and of per capita consumption by using the natural log. 3. If y is related to x by y = x*, how will the rates of growth ry and r,, be related? 4. Prove that if y = u/v, where u = f(1) and v = 9(!), then the rate of growth of y will bery = fu - fv, as shown in (10.25). 5. The real income y is defined as the nominal income Y deflated by the price level P. How is ry (for real income) related to ry (for nominal income)? 6. Prove the rate-of-growth rule (10.27). 7. Given the demand function Q = k/ P^, where k and n are positive constants, find the point elasticity of demand &g by using (10.28) (cf. Exercise 8.1-4). 8. (a) Given y = wz, where w = g(x) and z= h(x), establish that Eyx = =wx + Fxx- (b) Given y = u/v, where u = G(x) and v = H(x), establish that yx = Eux - Evx. 9. Given y = f(x), show that the derivative d(log, y)/(log, x)-log to base b rather than e-also measures the point elasticity &yx- 10. Show that, if the demand for money Ma is a function of the national income Y = Y(!) and the interest rate i =: i((), the rate of growth of My can be expressed as a weighted sum of ry and fi, where the weights are the elasticities of My with respect to Y and i, respectively. 11. Given the production function Q = F(K, L), find a general expression for the rate of growth of. Q in terms of the rates of growth of K and L. 4. Prove that for any two scalars g and k (a) k(A + B) = KA +KB (b) (g + K)A = gA + KA (Note: To prove a result, you cannot use specific examples.) 5. For (a) through (d) find C = AB. (a) A = [12 14 20 5 8 = 2 (b) A = $ 7 7 11 ( C) A = 2 9 8 = 3 4 6 10 10 (0) A = $ 3 B = 11 3 2 9 (e) Find (@) C = AB, and (ii) D = BA, if A = 8 = [3 6 -2] 6. Prove that ( A + B)(C + D) = AC - AD + BC +. BD. 7. If the matrix A in Example 5 had all its four elements nonzero, would x'Ax still give a weighted sum of squares? Would the associative law still apply? 8. Name some situations or contexts where the notion of a weighted or unweighted sum of squares may be relevant