MHB Partial derivatives of the natural logs

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The discussion focuses on finding the partial derivatives of the function Q=(1/3)logeL+(2/3)logeK. The user initially calculates the partial derivatives as ∂Q/∂L = 1/(3L) and ∂Q/∂K = 2/(3L). However, a correction is suggested for the partial derivative with respect to K, stating it should be ∂Q/∂K = 2/(3K). The conversation highlights the importance of careful differentiation in logarithmic functions. Overall, the correct derivatives are essential for further analysis of the function.
claratanone
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Find the partial derivatives of the following function:

Q=(1/3)logeL+(2/3)logeK

Any help would be much appreciated!

Below is my working out so far:

\frac{\partial Q}{\partial L}= \frac{\frac{1}{3}}{L}

\frac{\partial Q}{\partial K}= \frac{\frac{2}{3}}{L}

Are these correct?
 
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I suspect it is just a typo, but you want:

$$\pd{Q}{K}=\frac{\frac{2}{3}}{K}=\frac{2}{3K}$$
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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