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Need help differentiating an expression. Answer is given just unsure of the steps.

Hello, this problem states to differentiate the expression with respect to q. I am not sure of the steps taken to get the final result.

This is an economics problem differentiating the variable cost function to get the marginal cost function.

Re: Need help differentiating an expression. Answer is given just unsure of the steps

So you want to differentiate:

$\displaystyle 6(\frac{q}{10k^{0.4}})^{1.67}$ with respect to q?

In which case, we have:

$\displaystyle 6\frac{q^{1.67}}{(10k^{0.4})^{1.67}}$

=$\displaystyle \frac{6q^{1.67}}{(10k^{0.4})^{1.67}}$

But the $\displaystyle \frac{6}{(10k^{0.4})^{1.67}}$(everything except the $\displaystyle q^{1.67}$) is just constant. It's a stable value, like 4 or 7 - it isn't a variable.

For the sake of simplicity, I'm going to let $\displaystyle \frac{6}{(10k^{0.4})^{1.67}}=c$

When we differentiate $\displaystyle 4x^n$ with respect to $\displaystyle x$, we get: $\displaystyle 4nx^{n-1}$.

When we differentiate $\displaystyle ax^n$ with respect to $\displaystyle x$, we get: $\displaystyle anx^{n-1}$

We have: $\displaystyle c\times~q^{1.67}$

So, differentiating, we get: $\displaystyle 1.67c\times~q^{0.67}$

Then, slipping c back in as $\displaystyle \frac{6}{(10k^{0.4})^{1.67}}$, we have:

$\displaystyle \frac{1.67\times~6\times~q^{0.67}}{(10k^{0.4})^{1. 67}}$

and $\displaystyle 1.67\times~6=10.02$ which is where that number arises.