Answer:
The marginal profit is [tex]\frac{1000(100-x^{2})}{(x^{2}+100)^{2}}[/tex].
Step-by-step explanation:
The profit function p (x) is the difference between the revenue and cost function.
The profit function is given as follows:
[tex]p (x) = \frac{1000x}{100+x^{2}}[/tex]
Determine the marginal profit function as follows:
[tex]\text{Marginal Profit}=\frac{\text{d}}{\text{dx}} p (x)[/tex]
[tex]={\tfrac{\mathrm{d}}{\mathrm{d}x}\left[\dfrac{1000x}{x^2+100}\right]}}\\\\={1000\cdot{\tfrac{\mathrm{d}}{\mathrm{d}x}\left[\dfrac{x}{x^2+100}\right]}}}}\\\\[/tex]
[tex]=1000\cdot\frac{\frac{\text{d}}{\text{dx}}(x)\cdot (x^{2}+100)+x\cdot \frac{\text{d}}{\text{dx}} (x^{2}+100)}{ (x^{2}+100)^{2}}\\\\=\frac{1000(x^{2}-2x^{2}+100)}{(x^{2}+100)^{2}}\\\\=\frac{1000(100-x^{2})}{(x^{2}+100)^{2}}[/tex]
Thus, the marginal profit is [tex]\frac{1000(100-x^{2})}{(x^{2}+100)^{2}}[/tex].
