# 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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### Convex Functions Gradient Inequality : $f(x) \geq f(y) + \nabla{f(y)} \cdot (x-y)$

How do I prove that for a multivariate convex function $f:C\rightarrow \mathbb{R}$. Where $C$ is a convex set $f(x) \geq f(y) + \nabla{f(y)} \cdot (x-y)$ $\forall x,y \in C$
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### A High School Calculus inequality

The problem, encountered in a high school math textbook in the exercises on the MVT,goes as following: Let $f(x)=e^x-ex, x\geq 0$ and $f(\ln{2})<2$, prove that $y=(1-e)x+1$ is tangent to the graph ...
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### derivative of the sum

My book says: given $$M_j\equiv e^{x_{jt}\beta-\alpha p_{jt}}+\xi_{jt}$$ $$s_{jt}=\frac{M_j}{1+\sum_{k=1}^{50}M_k}$$ then \frac{...
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### How do I analytically check if a point is a local extremum when $f''(x)=0$?

If $f'(x_0)=0$ and $f''(x_0)=0$, how do I check whether or not it is a local extremum? The usual surefire method is to check the sign of $f'$ before and after $x_0$ , but I'm trying to prove a general ...
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### Is a differentiable multivariable function with continuous derivatives on analytic paths continuously differentiable?

Let $f: \mathbb R^n \rightarrow \mathbb R$ be an everywhere differentiable function; and continuously differentiable when restricted to any analytic path. Is then $f$ continuously differentiable? I ...
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### Area under graph and integration of dx

Integrating $dx$ within the bounds of $x_1$ and $x_2$ would result in $x_2-x_1.$ How does this make sense in terms of area under the graph?
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### How to find the upper bound of an integral given the area

I am trying to do some decent programming for once, and it looks like math is unavoidable. I got pretty far with no math background, but now I'm stuck after days of trying to find an answer. I have ...
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### Left side derivative [duplicate]

Let $(a,b)$ be an open interval in $\mathbb{R}$. $f:(a,b)\rightarrow\mathbb{R}$ $z\in (a,b)$ and let $f$ be continuous in $z$. Let $f$ be differentiable over $(a,b)\setminus\{z\}$. My question is ...
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### Proof that derivative of a vector valued function is the derivative of its components [closed]

I just read about vector valued functions and their derivatives. It says that to compute the derivative of a vector function, just compute the derivative of its components. But i can't make myself ...
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### What does the derivative really mean?

I was introduced to calculus a few weeks ago, and while I can "solve" problems consisting of derivatives and integrals, I still do not truly understand what the derivative means. Here are ...
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### Are Saddle Points only Possible in Non-Convex Functions?

Informally, I have heard that the following are only possible in Non-Convex Functions: Saddle Points Local Minimums My Question: Is this in fact true - Can we mathematically prove that Saddle ...
We have a known, increasing function $p:[0,1]\rightarrow[0,1]$. We want to find conditions on functions $R_1(\cdot),R_0(\cdot)$, $R_1(\cdot)$ known to be strictly increasing and $R_0(\cdot)$ known to ...