# Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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### How to derive Marshallian demand function, indirect utility function and the expenditure function for cobb douglas utility function

I am having a problem in deriving marshallian demand function, indirect utility function and expenditure function from following cobb-douglas utility function, U(X,Y)=A.X^alpha.Y^beta
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### Interesting Integral including $\ln x$

I would like to evaluate this integral: $$\int_0^1 \frac{\sin(\ln(x))}{\ln(x)}\,dx$$ I tried a lot, started by integral uv formula [integration-by-parts?] and it went quite lengthy and never ending. ...
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### Increase in wind speed through derivatives.

The power (in kW) of a windmill is P = 0.35 * v ^ 3. Where v is the wind speed in m/s. We must use derivatives to calculate the increase of the power, if the wind speed increases from 19 m/s to 20 m/s....
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### What is the gradient of $x^T A\, x$ with respect to the matrix $A$?

I have seen many times that the gradient of $x^TA\,x$ with respect to $x$ is $2A\,x$. But how do you find its gradient with respect to $A$?
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### Concentration of a drug by its derivative

The concentration C (in mg / L) of a drug in the blood t minutes after it is administered is represented by the function $C(t)=- 0.016t^2 + 2.32t$. Calculate the drug withdrawal during the 100th ...
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### Leibniz rule for vector-valued functions

If $f \in C^\infty(\mathbb{R}^d, \mathbb{R})$ and $g \in C^\infty(\mathbb{R}^d,F)$ for some Fréchet space $F$, what are the derivatives of their pointwise product $fg$? I guess that it has to be D^n(...
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### Prove that if $f(x)$ is differentiable at $x_0$ and $n \in \mathbb{N}$ then $\lim_{n\to\infty} n[f(x_0+1/n)-f(x_0)]$ [closed]

I've just started to study differentiation and this problem really haunts me in my sleep, because I feel like I know what the solution is going to look like I am just not able to execute it. I think I ...
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### $\int_{0}^{1}f(x)g(x)=0 \implies f(x)=0 \ \forall x \in [0,1]$

Let $f:[0,1]\to \mathbb{R}$ be a continuous function. If $\int_{0}^{1}f(x)g(x)=0$ for all continuous functions $g(x)$, then $f(x)=0 \ \forall x \in [0,1]$. I would like to know if my proof holds, ...
### Find right and left sided derivative of $|2^x - 2|$ at $x = 1$
Find right and left sided derivative of $|2^x - 2|$ at $x = 1.$ Right sided: $\lim_{x\to1^{+}}\frac{2^x-2-0}{x-1}$ Left sided: $\lim_{x\to1^{-}}\frac{2-2^x-0}{x-1}$ I don't know how to continue to ...