Question

If the total cost function for a manufacturer is given by $C=\frac{5x^2}{\sqrt{(}{x}^2+3)}+5000$, find marginal cost function.

• A. $\frac{3x(x^2+6)}{(x^2+3{)}^{\left(3/2\right)}}$
• B. $\frac{4x(x^2+6)}{(x^2+3{)}^{\left(3/2\right)}}$
• C. $\frac{5x(x^2+6)}{(x^2+3{)}^{\left(3/2\right)}}$
• D. None of these

Answer 1

Answer: (C)

$\frac{5x(x^2+6)}{(x^2+3{)}^{\left(3/2\right)}}$

Given cost function $C\left(x\right)=\frac{5x^2}{\sqrt{(x^2+3)}}+5000$

Marginal cost function is given by $C^{\prime }\left(x\right)$

$C^{\prime }\left(x\right)=\frac{\left(\sqrt{x^2+3}\right)\left(10x\right)-\frac{5x^2}{2\sqrt{x^2+3}}\times \left(2x+10\right)}{{\left(\sqrt{x^2+3}\right)}^2}$

$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}$

$C^{\prime }\left(x\right)=\frac{2(x^2+3)\left(10x\right)-5x^2\left(2x\right)}{2{\left(\sqrt{x^2+3}\right)}^3}=\frac{5x(x^2+6)}{{(x^2+3)}^{\frac{3}{2}}}$