# Tagged Questions

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

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### Second derivative with implicit differentiation

Question: Determine whether the given relation is an implicit solution to the give differential equation. Assume that the relationship does define y implicitly as a function of x and use implicit ...
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### Only once differentiable

Is there any example of a real function that is one-time-only differentiable, meaning there is $f'(x)$, but no $f''(x)$? I haven't been able to find any example... Of course it would be preferred if f ...
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### To determine the existence and the value of $\lim_{x \to 0} \frac {2^x-1} x$

I would like to somehow firstly show that $$\lim_{x \to 0} \frac {2^x-1} x$$ exists and determine the value of the limit. My first ideas were by Monotone Convergence. I have been able to prove that ...
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### The supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ when $x+y=2n$ for some fixed $n\in \mathbb N$

Let $S$ be the set of all tuples $(x,y)$ such that $x+y=2n$ for a fixed $n\in \mathbb N$. Then what is the supremum value of $x^{2}y^{2}(x^{2}+y^{2})$ $?$ I substituted $y=2n-x$ ...
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### How to differentiate $\sum_{t=0}^{T}$ problem?

Define $F(C, H)=\sum_{t=0}^{T}\beta^t (logC_t+\mu log H_t)$ where $\beta$ and $\mu$ are constant. I expand this function like $F(C, H)=\beta^0 (logC_0+\mu logH_0)+ \beta^1(logC_1+\mu logH_1)+\dots$ ...
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### why the integral of $\frac{dy}{y} =\ln(y)$?

I mean if I differentiate $\ln(y)$ the result will be $\frac{dy}{y}$ ? . What I know the diffential of $\ln(x)$ = $\frac{1}{x}$ right?. And following this idea what is going to happen if we ...
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How red-circled function with 1/D is equal to green-circled? Note: D is equal to dy/dx. Update: Complete pic
This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
### Finding the Derivative of $\sqrt{x}$
How can I find the derivative of $\sqrt{x}$ using first principle. Specifically I'm having difficulty expanding $\sqrt{x + h}$ or rather $(x + h)^.5$. Is there any generalized formula for the ...