# Tagged Questions

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

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### How to tell if a function has a cusp without a graph?

For my calculus exam, I need to be able to identify if a function is indifferentiable at any point without a graph. I thought this would be rather simple, but I messed up on the question x^(2/3) ...
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### Equality of mixed partials proof

I'm trying to prove the equality of mixed partials. My book has a proof but it's only for functions $\Bbb R^2 \to \Bbb R$ (and then that can be extended to $\Bbb R^2\to \Bbb R^n$ by applying the ...
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### Fixed point, bounded derivative

Let $p\in\mathbb{N}$. Let $f:I\to\mathbb{R}$ differentiable in the closed interval $I$ (bounded or not), with $f(I) \subset I$, and let $g = f\circ f\circ \cdots \circ f = f^p$, where $\circ$ means ...
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### Multivariable implicit function - Jacobi Matrix

Find the derivate $f',f''$ of the implicit function $z=f(x,y)$ defined by the following equation: $$F(x,y,z)=x^2+y^2+z^2-a^2=0$$ So the first step to build the Jacobi-Matrix $f'$ lead me to ...
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### How Can solve a n order Differential Equations

How can I solve the following equetion? what is the $$h(z).$$. $$z^n (z^n+1).|h'(z)|^n=const.$$.
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### Mean shift with Epanechnikov kernel

The multivariate Epanechnikov kernel is given by $$K_E(\vec{u}) = c(1-\vec{u})$$ if $\lVert u \rVert^2 \leq 1$ and $K_E(\vec{u}) = 0$ otherwise. When applying the mean shift algorithm, the update ...
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### If $f(x)$ has a vertical asymptote, does $f'(x)$ have one too?

So here is what I understand: If $f(x)$ is increasing/decreasing, then its derivative $f'(x)$ is positive/negative and... If $f(x)$ is increasing/decreasing, then the derivative of $f'(x)$ (...
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### Value of $V/(250\pi)$

A cylindrical container is to be made from certain solid material with the following constraints: It has fixed inner volume $V$ mm${}^3$ ,has a $2$ mm thick solid wall and is open at the top. The ...
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### Prove that $\overline{f(z)}$ is differentiable at $a \in D(0;1)$ if and only if $f'(a)=0$

Let $f$ be holomorphic in $D(0;1)$ and define $k$ by $k(z)=\overline{f(z)}$. Prove that $k$ is differentiable at $a\in D(0;1)$ if and only if $f'(a)=0$. What I tried was first, assuming $k$ is ...
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### Comprehension question about derivative in one point

Find the derivative of $f$ in $(x_{0} , y_{0})^{T}$ for: $$f(x,y)=\binom{x^4+2x^2y^2+y^4}{x^4+2x^2y^2+y^4}$$ Is it right to derivate $\partial x$ and $\partial y$ with $(x_0,y_0)^T$ ...
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### Differentiating the binomial coefficient

I took a lecture in combinatorics this semester and the professor did the following step in a proof: He showed that function $f: x \mapsto \binom{x}{r}$ is convex for $x > r - 1$ (in order to use ...
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### If $f(x) = x\log2,$ then find $f'(x)?$

I have a function (natural log): $$f(x) = x\log2$$ My textbook shows that the derivative of it is: $$f'(x)=\frac{x}{2}$$ But My teacher told me that we should take the derivative of whatever behind ...
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### How to solve $\lim_{x \to 0^+} \frac{x^x - 1}{\ln(x) + x - 1}$ using L'Hôpital

How could I solve \begin{align*} \lim_{x \to 0^+} \frac{x^x - 1}{\ln(x) + x - 1} \end{align*} using L'hôpital? Analysing the limit we have $0^0$ on the numerator (which would require using ...
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### Change of variable problem. [closed]

If I have $\tilde{u} = \Omega(x, u)$ and $\tilde{x} = \Gamma(x,u)$ then how I can prove $\tilde{u_{\tilde{x}}} = (\Omega_{u}u_{x}+\Omega_{x})(x_{\tilde{x}}+x_{\tilde{u}}\tilde{u_{\tilde{x}}})$
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### Can the second derivative of a function be interpreted as the slope of its “concavity lines”?

Can the second derivative of a function be interpreted as the slope of its "concavity lines"? For example consider the following picture: Does $f''$ for each point $x$ that corresponds to an arrow ...
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### Matrix derivative (chain rule application)

Let $x$, $y$ by vectors s.t. $x=f(y)$ and let $B$ be a constant matrix. What is $\frac{\partial x'Bx}{\partial y}$? The partial derivative $\frac{\partial x'Bx}{\partial x}=2Bx$ and we need to use ...
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### Discuss the continuity and differentiablity of given function.

If $\big[\cdot\big]$ denotes floor function (i.e the integral part of $x$) and $$f(x)=\big[x \big] \left(\frac{\sin \frac{\pi}{\big[x+3\big]}+\sin \pi \big[x+3\big]}{3+\big[x \big]} \right)$$, then ...
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### exercice analysis; inverse theorem, implicit function theorem, locally immersions and submersions, post theorem

Could anyone help me find lists of exercises (in books or other materials) analysis in R for a qualification examination. Threads 0) differentiability in R 1) the inverse function theorem 2) implicit ...
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### About $\frac{\partial^n f}{\partial x^n}$ ,for $f(x)$,What should I think, when $n[\in (\mathbb R\backslash \mathbb Q)^+]$ or $(\in \mathbb C)$? [duplicate]

About $\frac{\partial^n f}{\partial x^n}$ ,for $f(x)$,What should I think, when $n[\in (\mathbb R\backslash \mathbb Q)^+]$ ,pozitive irrational? or $(\in \mathbb C)$ For example; $f(x)=x^2+3x\quad$ ...
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### A new “differential” form for the antiderivative?

The derivative is in general notated by: $\frac {dy}{dx} = \frac d{dx} f(x)$ It has come to my understanding quite recently that dx and dy are actual quantities and not just notational garbage. So ...
### Show that $\text{rank}(Df)(A) = \frac{n(n+1)}{2}$ for all $A$ such that $A^TA = I_n$
We identify $\mathbb R^{n \times n}$ with $\mathbb R^{n^2}$ and define $f:\mathbb R^{n^2} \to \mathbb R^{n^2}, A \mapsto A^TA$. Show that $\text{rank}(Df)(A) = \frac{n(n+1)}{2}$ for all $A$ such ...