# 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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### Show that $\frac{x^3+y^3}{x^2+y^2}$ is not differentiable at $(0,0)$

Let be $f:\mathbb{R}^2\to\mathbb{R}$ where $$f(x,y):=\begin{cases} \frac{x^3+y^3}{x^2+y^2},&(x,y)\neq (0,0)\\0,&(x,y)=(0,0).\end{cases}$$ Show that $f$ is not differentiable at $(0,0)$. My ...
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### Taylor Approximation of Third Order

Problem: Given the function $f(x, y) = e^{x^2} \log(1 + x + y)$ near the point $(0, 0)$, find the third-order Taylor approximation of the function at $(0, 0)$ using known series and verify the ...
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### Let f : R → R be differentiable at x = 0. Define g(x) = f(x^2 ). Show that g is differentiable at 0 (by the chain rule). [closed]

Let $f : \mathbb{R} → \mathbb{R}$ be differentiable at $x = 0$. Define $g(x) = f(x^2 )$. Show that $g$ is differentiable at $0$(by the chain rule).
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### Using the Airy Function to give a solution to a differential equation

Questions (b) to (e) attached How do I solve part (e)? It says to use the Airy Function and its derivative to give a solution to the differential equation but I am really unsure as how to proceed. ...
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### Justify if there exist a differentiable function f such that $|f(x)|<2$ and $f(x)f'(x)\geq \sin(x)$

The problem states: Justify if there exist a differentiable function f such that $|f(x)|<2$ and $f(x)f'(x)\geq \sin(x)$ for every $x\in \mathbb{R}$. What I got by another question I made about a ...
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### How to find the derivative of Q function [closed]

How to take the derivative of the given expression with respect to $x$ $$Q \left( \frac{\sqrt L ( \ln (1+x) - R)} {\sqrt {1- \frac{1}{( 1+x)^2}}} \right)$$ where $Q(x)$ is the complementary error ...
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### Justification for differentiation under integral

Given a function I seek to find its derivative $$f(x) = \int_{\frac{1}{x}}^{\frac{e^x}{x}} \frac{\cos(xt)}{t} \, dt, \quad (x>0)$$ My question is regarding the justification of the differentiation ...
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### Comparing two series expressions for $1/\zeta(s)$. What can be said about their complex roots?

The following two expressions involving the inverted Riemann $\zeta(s)$ functions are well known: \begin{align} \frac{1}{\zeta(s)} &= \sum_{n=1}^\infty \frac{\mu(n)}{n^s} \\ -\frac{\zeta'(s)}{\...
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### Ambiguity in solving differential equations

Suppose we want to solve the differential equation $y'=x \sqrt{y}$. Easy right? Because you can transform the equation into a separable one. However, I think that there are more than meets the eye. ...
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### Calculating the Derivative of a Complex Vector Function

I am trying to calculate the derivative of the function $$f(x) = x^T a (x^T a)^* = x^T a x^H a^* = x^T a a^H x^*$$, where $(.)^T$, $(.)^H$, and $(.)^*$ represent the transpose, Hermitian (conjugate ...
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### The upper bound of a definite integral involving a derivative of an exponential function

I have a question. Is there anyone who knows how to evaluate the upper bound of the absolute value of the following integral. $$\int_{n}^{\infty} \Bigl(\frac{\exp(iax)}{x}\Bigr)^{(n)} \exp(ix) dx$$, ...
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### I can't show that for a function f:R^2-->R if df/dx=df/dy and f(x,0)>0 then f(x,y)>0. [closed]

I can't show that for a function f:R^2-->R if df/dx=df/dy, f/in C^1 and f(x,0)>0 for all x in R then f(x,y)>0.
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We are supposed to test the validity of two statements: (1) There exists a differentiable function $g:R\rightarrow R$ such that $g(x^3+x^5)=e^x-100$ (2) There exists a continuous function $g:R\... 0 votes 0 answers 50 views ### Equivalence of Taylor Expansion and L'Hospital Is the rule of L'Hospital equivalent to Taylor Expansion?: A analytic (or for this sake one time differentiable) function$f(x)$can be expanened around$x_0$with$f(x) \approx f(x_0) + f'(x_0) (x-...
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Consider a function $f(x,y)$. Let $x, y \geq 0$, and $\frac{\mathrm d}{\mathrm dx}f(x,y), \frac{\mathrm d}{\mathrm dy}f(x,y) >0 \ \forall x, y$. I would tend to think that the cross partial ...
I've been stuck on this problem and I honestly don't know where to start. How much variation dr in the radius of a coin can be tolerated if the volume of the coin is to be within $11000$ of their ...