Chapter 7

Use Jacobian determinants to test the existence of functional dependence between the paired functions. (a) \(y_{1}=3 x_{1}^{2}+x_{2}\) \(y_{2}=9 x_{1}^{4}+6 x_{1}^{2}\left(x_{2}+4\right)+x_{2}\left(x_{2}+8\right)+12\) (b) \(y_{1}=3 x_{1}^{2}+2 x_{2}^{2}\) \(y_{2}=5 x_{1}+1\)

Given the total-cost function \(C=Q^{3}-5 Q^{2}+12 Q+75\), write out a variable- cost (VC) function. Find the derivative of the VC function, and interpret the economic meaning of that derivative.

Given \(y=u^{3}+2 u,\) where \(u=5-x^{2},\) find \(d y / d x\) by the chain rule.

Find the derivative of each of the following functions: $$(a) y=x^{12}$$ $$(b) y=63$$ $$(c) y=7 x^{5}$$ $$(d) w=3 u^{-1}$$ $$(e) w=-4 u^{1 / 2}$$ $$(f) w=4 u^{1 / 4}$$

Find \(\partial y / \partial x_{1}\) and \(\partial y / \partial x_{2}\) for each of the following functions: \((a) y=2 x_{1}^{3}-11 x_{1}^{2} x_{2}+3 x_{2}^{2}\) (b) \(y=7 x_{1}+6 x_{1} x_{2}^{2}-9 x_{2}^{3}\) (c) \(y=\left(2 x_{1}+3\right)\left(x_{2}-2\right)\) \((d) y=\left(5 x_{1}+3\right) /\left(x_{2}-2\right)\)

Given \(w=a y^{2}\) and \(y=b x^{2}+c x,\) find \(d w / d x\) by the chain rule.

Find \(f_{x}\) and \(f_{y}\) from the following: \((a) f(x, y)=x^{2}+5 x y-y^{3}\) (b) \(f(x, y)=\left(x^{2}-3 y\right)(x-2)\) \((c) f(x, y)=\frac{2 x-3 y}{x+y}\) \((d) f(x, y)=\frac{x^{2}-1}{x y}\)

Find \(f^{\prime}(1)\) and \(f^{\prime}(2)\) from the following functions: $$(a) y=f(x)=18 x$$ $$(b) y=f(x)=c x^{3}$$ $$(c) f(x)=-5 x^{-2}$$ $$(d) f(x)=\frac{3}{4} x^{4 / 3}$$ $$(e) f(w)=6 w^{1 / 3}$$ $$(f) f(w)=-3 w^{-1 / 6}$$

Use the chain rule to find \(d y / d x\) for the following: (a) \(y=\left(3 x^{2}-13\right)^{3}\) (b) \(y=\left(7 x^{3}-5\right)^{9}\) (c) \(y=(a x+b)^{5}\)

Differentiate the following by using the product rule: \((a)\left(9 x^{2}-2\right)(3 x-1)\) \((b)(3 x+10)\left(6 x^{2}-7 x\right)\) (c) \(x^{2}(4 x+6)\) \((d)(a x-b)\left(c x^{2}\right)\) \((e)(2-3 x)(1+x)(x+2)\) \((f)\left(x^{2}-3\right) x \cdot 1\)