# 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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### Let $f:[0,1] \to \mathbb{R}$ be continuously differentiable function

I was practicing calculus today and stumbled across this problem. I have tried solve using properties of inequalities but it doesn't get me far. Let $f:[0,1] \to \mathbb{R}$ be continuously ...
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### Getting value of x by differentiating a given equation

If we consider an equation $x=2x^2,$ we find that the values of $x$ that solve this equation are $0$ and $1/2$. Now, if we differentiate this equation on both sides with respect to $x,$ we get $1=4x.$ ...
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### Find parametric equations for the midpoint $P$ of the ladder

The following problem appears at MIT OCW Course 18.02 multivariable calculus. The top extremity of a ladder of length $L$ rests against a vertical wall, while the bottom is being pulled away. ...
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### Is the 1st definition about the differentiability of $f : A \to \mathbb{R}^m$ equivalent to the 2nd definition?

It is convenient to define a function $f :\mathbb{R}^n \to \mathbb{R}^m$ to be differentiable on $A$ if $f$ is differentiable at $a$ for each $a \in A$. If $f:A\to \mathbb{R}^m$, then $f$ is called ...
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### Directional derivative along the intersection of two surfaces

How can i find the intersection curve between these two surfaces $$\left\{ \begin{array}{cc} 2x^2 + 2y^2 − z^2 &= 50\\ x^2 + y^2 -z^2 &= 0 \end{array} \right.$$ I need it to find the ...
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### Can a tangent to a curve can also be its normal?

This question is related to applications of derivatives.I know the answers is yes but I can’t visualize the diagram
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### Derivative determinant function of a matrix

Basically this is the problem I am trying to solve, I differentiated it but am pretty sure it isn't correct, used the formula: $$\frac{f(I+h) - f(I)-Ah}{h}$$ where A is the derivative and I is the 2 ...
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### Complex derivative of Hadamard product inside Frobenius norm

I'm trying to find the complex derivative of $$||R - P \circ \gamma \gamma ^H||_F ^2$$. with respect to $\gamma$. I saw the post regarding the real counterpart of the same question here. However, ...
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### Central Difference Approximations

Hi Guys I was going through the different approximations which can be used for differentiation such as the forward difference, the backward difference and lastly the central difference approximations. ...
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### Differential Equation using finite difference method

I am working on the following question $$y''+8(\sin^2 \pi y) y=0$$ where the initial conditions are $$y(0) = y(1) = 1$$ Now by the finite difference method i have made the substitution for $y''$ ...
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### Compute the derivative of the determinant function on 2x2 matrices

Using the definition of the derivative and limit, compute the derivative of the determinant function on 2x2 matrices at the identity (Which we consider as a subset of $\mathbb{R}^4$ under the ...
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### rate of change in natural log $\ln(x)$

I have confusion with the rate of change in natural logarithm, as we know that, in analytical manner \begin{gather*} y\ =\ \ln( x)\\ \\ \frac{dy}{dx} \ =\ \frac{1}{x}\\ \end{gather*} or in a ...
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### Derivative of a unit vector

Consider a vector function $r: \mathbb{R} \to \mathbb{R}^n$ defined by $r(t)$. We use $\hat{r}$ to denote its normalized vector, and $\dot{r}$ to denote $\frac{d}{dt}r(t)$. We know that the derivative ...
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### Caratheodory differentiablity vs. continuously differentiable

Here is the definition of Caratheodory differentiability: Let $I\subseteq \mathbb{R}$ be an open interval, $a\in I$, $f:I\to\mathbb{R}$. We say $f$ is Caratheodory differentiable at $a$ if there ...
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### Sum rule for infinite sums

Why does the sum rule of differentiation fail sometimes for an infinite sum.(In terms of the derivation of the sum rule.)An example where it fails is here: Interchanging the order of differentiation ...
I have the following Problem: Let $g \in C^1(\mathbb{R^2};\mathbb{R})$. Show that an injective $f \in C^1 ((-1,1);\mathbb{R^2})$ exists, so $g \circ f$ is constant. The hint asks if there is a (x,... 0answers 35 views ### Definition of differential equations [closed] If we consider the LDE to be of the form $$a_n(t)y^{(n)}(t)+a_{n-1}(t)y^{(n-1)}(t)+......+a_1(t)y^{(1)}(t)+a_0(t)y(t)=g(t)$$ and a non-LDE includes terms which have the dependent variable as ... 1answer 78 views ### Homework Assignment, Function Totally Differentiable, Correct Solution? We are given the following function: \begin{align*} f: \mathbb{R}^2 &\rightarrow \mathbb{R} \\ f(x,y) &= \begin{cases} {x^3y^4 \over x^6 + y^4} \ & (x,y) \neq (0,0) \\ 0 \ &(... 1answer 64 views ### Inverse of Laplace transform of\mathscr{L}\{f(t)\}=\frac{1}{s^{2}}\tanh\left(\frac{s}{2}\right)$I can't find its inverse transform, I had thought in$$f(t)=\begin{cases} t \space\space\space\space\space\space\space\space\space\space\space\text{ if }0 \leq t<1,\\ -t+2 \space\text{ if }1\leq t&... 1answer 17 views ### Uniqueness of the Frechet Derivative: the role of$x \in int_X(T)$I'm currently trying to learn some functional analysis as a way to improve my ability to read economic theory papers. I've come across what I thought was a simple proof but on reflection I don't think ... 0answers 33 views ### Interchange of derivatives and integrals I am trying to justify the following exchange of derivatives and integrals. Suppose that$s, \theta \in [a, b] \subset \mathbb{R}$. Let$\pi_\theta(s) = \max [0, \theta - s]$, so it is continuous and ... 1answer 45 views ### Why isn't$f(x)=0$ever mentioned as a solution to$f'(x)=f(x)$? I know that$f(x)=e^x$is the accepted and useful solution to$f'(x)=f(x)$, but why isn't$f(x)=0$ever mentioned as a solution as well? Is it simply because it's not useful? 0answers 23 views ### Definition of slope of parametric tangent Let$x = u(t)$and$y = v(t)$be a parametric representation of a curve. If we assume$u$is one-one on some open interval in$t$, we can define$f = v \circ u^{-1}$. Then the curve$(u(t), v(t))$on ... 3answers 47 views ### Show that not exists any polynomial function such that$f(x) = \log (1+x)$. [duplicate] Does anyone have any idea on that problem? Let$f : \mathbb{R} \to \mathbb{R}$be a polynomial function. Show that not exists any$f$such that$f(x) = \log (1+x)$. It's easy to show that$a_0 = 0$... 0answers 29 views ### Help with differentiability proof for piecewise function I was able to prove that$f(x)$is continuous only when$x$is$0$or irrational. But I'm unable to prove that it is not differentiable at any point (which is what the book's answer says). Can someone ... 1answer 87 views ### Why can$e^x$be defined as the unique function$f(x)$such that$f(x)=f'(x)$and$f(0)=1$? The definition that$e^x$is the unique function$f(x)$such that$f(x)=f'(x)$and$f(0)=1$has two problems for me: How is$e^x$the unique function that satisfies this property?$ke^x$also has ... 0answers 28 views ### Applicability of L'Hôpital's rule The derivation of L'Hôpital's rule requires Cauchy's theorem, which, in turn, requires the following conditions: two functions$f(x)$and$g(x)$must be continuous and differentiable in an interval$[...
I am interested if there is geometric meaning (using graphs) of $(1 + \frac{1}{n})^n$ when $n \rightarrow \infty$. Also, is there visual explanation of why is $e^x = (1 + \frac{x}{n})^n$ when \$n \...