# Questions tagged [derivatives]

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

20,871 questions
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### Taking derivative with chain rule

Suppose I have a function: $f(x(\eta),\eta)$ and I want to take the derivative with respect to $\eta$. Note that $f$ is a function of $x$ and $\eta$ and that $x$ itself is a function of $\eta$. I am ...
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### property of the tangent line.

If the traditional way to define the tangent line to a curve $f(x)$ through the point say $(a , f(a))$ is: ( the tangent line through the point $(a ,f(a))$ is the line that passes through this point ...
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### Application of Taylor's theorem: find upper bound for remainder?

Suppose $f$ is a $C^2$ function with compact support. I.e. $f$ is $0$ outside a closed interval. Then $f,f',f''$ are uniformly continuous and bounded on $\mathbb{R}$. My textbook then claims that the ...
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### How to find a partial derivative in order to check whether the function is differentiable

I need to find out whether the following function is differentiable at the point $(0,0)$. $$f(x)=\sqrt[3]{1+|x|^{2/e}\cdot|y|^{3/\pi}}$$ I think I need to find the partial derivatives first, but the ...
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### Is integral of a function differentiable?

If we have a continuous function $f(x)$ and its integral is $F(x)=\displaystyle \int_a^x f(x)\ dx$, will $F(x)$ be differentiable?
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### Complex differentiable function with non-continuous partial dervative [on hold]

I'm looking for a complex-valued function $f$ which is complex differentiable in $z_0$ but where the partial-derivatives are non-continuous in $z_0$. Can someone give an example? Best! Annette
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### Is a function differentiable at a point if its derivative is continuous at that point?

My professor said that the title statement might not always be the case and gave $$x^2 \sin\left(\frac{1}{x}\right)$$ at $x=0$ as a counter-example. But I don't seem to understand its ...
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### Matrix derivative in images matching problem

Problem Suppose zero-centered matrices $\mathbf{X}$ and $\mathbf{Y}$ of shape $\mathbb{R}^{n\times 2}$. Each row of $\mathbf{X}$ and $\mathbf{Y}$ represents a point on 2-D plane. Therefore, they each ...
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### How to create a cubic spline between the lines x=0 and y=1?

I am trying to create a simple cubic spline from points (0,0) to (m,1) connecting the lines y=1 and x=0. However, I am having trouble getting the spline to be tangential to the x=0 line at (0,0). ...
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### How to Use Differentials to Estimate the Percentage Change in $r$, if $x$ increases by 6%. Let $r=6x^{-1/6}, x>0$

I am trying to determine how to use differentials to estimate the percentage change in $r$, if $x$ increases by 6%. Let $r=6x^{-1/6}, x>0$. So far, I have done the following steps: 1) Determine ...
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### Dual via KKT for convex problem

I would appreciate a bit of help in order to continue simplifying the following problem. I have a primal convex problem $\mathcal{P}$ and I am trying to find its dual $\mathcal{D}$. Due to the ...
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### Check of $f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$ properties

For function defined as $$f(x)=\sum_{n=1}^{\infty}\frac{1}{x^2+n^2}$$ check if $f$ is continuous and differentiable function. My approach: I would like to use the connection between this sum and ...
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### Is my solution of $\frac{d }{dx}\int_0^{\cos x}\sqrt{1+t^4}dt$ correct?

$$\frac{d }{dx}\int_0^{\cos x}\sqrt{1+t^4}dt$$ $$\frac{(\sqrt{1+\cos^4x}-1)dx}{dx}$$ $$\sqrt{1+\cos^4x}-1$$ The answer seems weird to me, but I see no other way to do this. Was this correct? If not, ...
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### $f(x)=(x-a)(x-a_2)…(x-a_n)\in F[x]$ where $F$ is a field and $a_j\in$ for $j=1,2,…,n$ has no repeated roots iff gcd$(f(x),f'(x))=1\in F[x]$
This makes sense to me if $a_j\ne a_k$ for $j\ne k$ as $(x-a_j)=0 \implies a_j$ is a root of $f(x)$. So if all $a_j$ are different, then all the roots will be different. Do I have to somehow show this ...