Question

1. Derivatives practice: a. Find the marginal product of labor when F(x)-4KL (Take the derivative with respect to L b. Find the marginal product of capital when F(x)-4KL (Take the derivative with respect to Find the marginal product of labor when F(x) = 61K3 (Take the derivative with respect to L). c. d. Find the marginal product of labor when F(x) 61/2 K1/ (Take the derivative with e. Find the marginal product of labor when F(x) = 11/4K3/4 (Take the derivative with f. Find the marginal product of labor when F(x)-LV4K3/4(Take the derivative with respect to L) respect to L) respect to K). Find the marginal product of capital when F(x)=61K3 (Take the derivative with respect to K). g. Find the marginal product of capital when F(x)-= 61 + K"(Take the derivative with respect to K). h. i. L3 (Take the derivative with Find the marginal product of capital when F(x)--6K + respect to K).

Answer 1

1. F = 4KL

Differentiating F wrt L, we get

$\large \frac{dF}{dL} = \frac{d(4KL)}{dL}$

$\large MP_L = 4K$

Marginal product of Labour = 4K

2. F = 4KL

Differentiating F wrt K, we get

$\large \frac{dF}{dK} = \frac{d(4KL)}{dK}$

$\large MP_K = 4L$

Marginal product of capital = 4L

3. $\large F = 6LK^3$

Differentiating F wrt L we get

$\large \frac{dF}{dL} = \frac{d(6LK^3)}{dL}$

$\large MP_L = 18K^2$

4. $\large F = 6K^{\frac{1}{2}}L^{\frac{1}{2}}$

Differentiating F wrt L we get

$\large \frac{dF}{dL} = \frac{d(6L^{\frac{1}{2}}K^{\frac{1}{2}})}{dL}$

$\large MP_L= \frac{1}{2}*{6K^{\frac{1}{2}}L^{\frac{1}{2}-1}}$

$\large MP_L= 3K^{\frac{1}{2}}L^{\frac{-1}{2}}$

$\large MP_L= \frac{3K^{\frac{1}{2}}}{L^{\frac{1}{2}}}$

5. $\large F = L^{\frac{1}{4}}K^{\frac{3}{4}}$

Differentiating wrt L we get

$\large \frac{dF}{dL} =\frac{d( L^{\frac{1}{4}}K^{\frac{3}{4}})}{dL}$

$\large MP_L=K^{3/4}\frac{d L^{\frac{1}{4}}}{dL}$

$\large MP_L=\frac{1}{4}K^{3/4} L^{\frac{1}{4}-1}$

$\large MP_L=\frac{1}{4}K^{3/4} L^{\frac{-3}{4}}$

$\large MP_L=\frac{K^{3/4}}{4L^{3/4}}$

6. $\large F = L^{\frac{1}{4}}K^{\frac{3}{4}}$

Differentiating F wrt K we get

$\large \frac{dF}{dK} =\frac{d( L^{\frac{1}{4}}K^{\frac{3}{4}})}{dK}$

$\large MP_K=L^{1/4}\frac{d K^{\frac{3}{4}}}{dL}$

$\large MP_K=L^{1/4}\frac{3}{4K^{\frac{1}{4}}}$

7. $\large F = 6LK^3$

Differentiating F wrt K, we get

$\large MP_K= 18LK^2$

8. $\large F = 6L + K^3$

Differentiating F Wrt to K we get

$\large MP_K= 2K^2$

9. $\large F = 6K + L^3$

Differentiating wrt K we get

$\large MP_K= 6$

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