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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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Interpretation of $\nabla$ operator in an expression

Let a tensor (3x3) be of the form $U = \mathbf{u}\mathbf{v}$ ($\mathbf{u}$ and $\mathbf{v}$) being two fluid velocity vectors (of dimension 1x3). In my analysis, for such a tensor $U$, following ...
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1answer
18 views

Finding the derivative of inverse of the product of matrices

I need to calculate the following derivative of the product of several matrices (one of which is the inverse of a product of matrices) with respect to one of the matrices in question: $$\frac{\delta(\...
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1answer
25 views

Can you differentiate the length of a function?

I came across a tricky issue while setting up an optimization problem. Is it possible to differentiate $len^2(f(x)_B)$, the squared length of some function $f(x)$ under some basis $B$? I know this ...
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Why 2 dimensional gradient's partial derivative determines direction in xy plane

I was reading about gradient and I still can grasp why the components of the gradient are partial derivative. To be more precise, let say for a 2 dimensional function that for the point (5,3) my ...
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what is the partial derivative of $tr[\rho\log(\sum_xp_x\rho_x)]$ with respect to $p_x$? [on hold]

anyone knows the how to get $\frac{\partial}{\partial p_x}tr[\rho\log(\sum_xp_x\rho_x)]$? The "tr" means trace and $\rho,\rho_x$ are positive semi-definite matrices.
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2answers
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Understanding how integration by parts is done in Gamma function

The the Gamma function is defined as... . I'm looking into how the Gauss representation of the Gamma function is derived and the first step is integration by parts. No steps are shown and the ...
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2answers
14 views

Show that $\Lambda(t)\le-\frac C2\Lambda'(t)$ if and only if $e^{2t/C}\Lambda(t)$ is nonincreasing

Let $\Lambda\in C^1([0,\infty))$ and $C>0$. Why does $$\Lambda(t)\le-\frac C2\Lambda'(t)\;\;\;\text{for all }t>0$$ hold if and only if $e^{2t/C}\Lambda(t)$ is nonincreasing in $t$? Is this just ...
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1answer
28 views

What does my textbook mean by Hypothesis here?

I am reading through page 278 of the textbook "Calculus: Early Transcendentals" 8th Edition, by James Stewart. I am confused by his use of the word "hypothesis" here. I thought a hypothesis was ...
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1answer
16 views

Differentiability implies Lipschitz continuity (multivariable)

I am studying from Marsden: Elementary Classical Analysis ($2^{\rm{nd}}$ ed.). I am not able to write down the complete proof of the following theorem (Theorem 6.3.1, page 334). The theorem ...
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15 views

Jacobian for a semi linear differential equation problem

How would I find the jacobian in this case? Normally the Jacobian is calculated using partial derivatives of F with respect to each of the variables, but since we are using the centered finite ...
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1answer
33 views

If $g'(0)>0$ write this expression $\lim\limits_{x \to 0} \frac{\sin(f(x))}{\sin(g(x))}$ using $f(0),f'(0)$ and $g(0)$.

If $g'(0)>0$ write this expression $\lim\limits_{x \to 0} \frac{\sin(f(x))}{\sin(g(x))}$ using $f(0),f'(0)$ and $g(0)$. This came up in my Analysis 1 exam, and i couldn't do it.
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1answer
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Calculus - Integral - existence problem

We are given a function $f$, $f$ is integrable (in the riemann sense) in $[a,b]$ and also $f'$ is a continuous, and $f(a)=f(b)=0$. Prove that there exists a point $c$ such that $|f'(c)| \geq \frac{4}{...
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33 views

Help with Limit 9

I have the following system of partial derivatives: $$\frac{\partial Y}{\partial K}=\frac{1}{K}\left ( Y-\frac{\partial Y}{\partial L}L \right )$$ $$\frac{\partial Y}{\partial L}=\alpha \left (\frac{...
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2answers
23 views

prove that if a function is differentiable then other function is constant

I got this exercise: Let $\sigma:R\rightarrow R$ be continuous function. And let: $f(x,y) = x\sigma(\frac{y}{x})$ if $x \ne 0$ $f(x,y) = 0$ if $x = 0$ Prove that if $f$ is differentiable in $(0,...
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3answers
55 views

How do I find the derivative of $x$ with respect to $y$ when $y = e^{-x^2}$?

When I have $y = e^{-x^2}$ with $-1 \leq x \leq 1$, I want to find the derivative of $x$ with respect to $y$. Can I take the log of both sides? $$\ln y = -x^2$$ From here, can I say that $-x^2$ is ...
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How do I find the derivative of $y = e^{-x^2}$ with respect to $y$?

I want to find the derivative of $y = e^{-x^2}$ with $-1 \leq x \leq 1$ Can I take the log of both sides? $$\ln y = -x^2$$ From here, can I say that $-x^2$ is always = $x^2$? If so, I get: $$\ln y =...
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1answer
32 views

When is the second derivative the same as the product of two first derivatives?

In the book Mathematical Methods for Physics and Engineering by RIley Hobson and Bence, an example is given in chapter 5 of using partial differentiation. The example problem Now the first partial ...
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1answer
22 views

For which value of $x$ is the average rate of change equal to the instantaneous rate of change?

The average rate of change for $f(x)=x^2+4x-6$ on the interval $[1,3]$ is $8$. I am not interested in final answer but more how to get there. I am going through calculus right now and already know ...
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1answer
49 views

Calculate $ dy\over dx $ at $\;x=e \;$ [on hold]

if $$\; \; x\ln(\ln x)-x^2+y^2=4 $$ $ $ and $ y>0 \;\; then \;$ $ dy\over dx $ at $\;x=e \;$ is equal to :
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5answers
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Find number of zeros $ x = \ln |x-a| $

Find number of zeros $$ x = \ln |x-a| $$ depending on the value of $a$. My try Usually I solve task like that in use of derivatives: Let $$f(x) = x - \ln |x-a| $$ Ok now it is time for derivatives ...
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0answers
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Derivative (Jacobian) of transposed function

Let $x \in R^n$, $F \in R^{m \times n}$ and $f(x) = Fx$. It's easy to conclude that the Jacobian of $f(x)$ is $Df(x) = F$. Where $Df(x)_{ij} = \frac{\partial f_i}{\partial x_j}$. Therefore $\nabla ...
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Product Rule for vector output functions

In Spivak's calculus of manifolds there is a product rule given as below. if $f,g:\mathbb{R}^n \rightarrow \mathbb{R}$ are differentiable at a, then $D(f*g)(a)=g(a)Df(a)+f(a)Dg(a).$ My question is, ...
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Are most physics books wrong about the covariant derivative and connection?

I have always read in many physics books that a valid way of intuitively introducing the covariant derivative and the connection was the following: (example in GR but same thing for gauge theories) ...
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Prove recursion relation on integrals [on hold]

With $$ I_n:=\int_{0}^{1}x^{n}\cos\left(\pi x \over 2\right)\,\mathrm{d}x\,, $$ prove $\displaystyle I_{n} + {8 \over \pi^{3}}n\left(n - 1\right)I_{n - 1} = {2 \over \pi}$ for $n\ge 2$.
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1answer
53 views

How can I know the derivability of this function?

I've been working on this problem: Study the differentiability of the following function $f$ at $x=0$: $$ f=\begin{cases} \dfrac{\cos(3x)e^{3x} - e^x}{\ln(1+x)} & \text{if} x>0\\ 2 &\text{...
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21 views

Implicit function derivative: Simplification

Simplify the following system of inequities: $$\frac{d^2y}{dx^2}<\frac{dy}{xdx}$$ $$\frac{d^2x}{dy^2}<\frac{dx}{ydy}$$ Known that $F(x,y)=0$. $F$ is monotonic. I got something like this: $$\...
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3answers
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Derivative of (1-t/m)^m

We know that... . However, I am trying to derive with respect to n... . But I am unsure how to do so as n is the exponent outside of the brackets as well as inside the brackets. The farthest I ...
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2answers
76 views

Why $x^2$ in $\frac{d^2y}{dx^2}$? [duplicate]

How can the part beneath in Prof Magidin's answer be demystified to a high-schooler? So you are trying to describe the change in "the-change-in-$y$", relative to how $x$ is changing. $x$ is only ...
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1answer
40 views

How to differentiate $E[X\vert Y\geq y](1-F(y))$ w.r.t $y$

It is known that $$E[X\vert Y\geq y](1-F(y)) = \int_y^\infty E[X\vert Y=t]f(t)dt$$ To take the derivative, I think this is just Leibniz rule, but I don't know how to handle the conditional ...
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1answer
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How partial derivatives in this answer were calculated, and how to understand a notation in the answer

There is a question asking how to calculate $\frac{\partial \tilde{x}}{\partial a}$, where $\tilde{x}$ is defined as follows. $$\mathbb{E} \left( x | x > a \right) = \frac{ \int_a^{\infty} x f\...
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2answers
58 views

Find the ratio between the radius and the height

A coffee filter has the shape of an inverted cone. Water drains out of the filter at a rate of 10 cm$^3$/min. When the depth of water in the cone is 8 cm, the depth is decreasing at 2 cm/min. What is ...
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2answers
40 views

Derivative of matrix exponential [on hold]

What is the derivative of $e^{(x-y)Q}$ with respect to $y$, where $x$ and $y$ are scalars and $Q$ is a transition rate matrix?
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1answer
48 views

Product Rule for $\mathbb{R}^n$

Let g: $\mathbb{R}^m$ → $\mathbb{R}^n$ differentiable and h: $\mathbb{R}^m$ → $\mathbb{R}$ differentiable with $x\in \mathbb{R}^m$. How to prove product rule that ($hg$)($x$)= $h'(x)g(x) + h(x)g'(x)$ ...
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0answers
19 views

A problem about the existence of partial derivative of composite funcitons

I was troubled in a problem. I need to consider: $$f: \mathbb R^m\to\mathbb R, g=(g_1,g_2,...,g_m):R^n\to\mathbb R^m.$$ The partial derivative of $g_i$ with respect to all variables at $x_0 (\in \...
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1answer
34 views

Absolutely continuous function not differentiable on uncountable set

From real analysis one knows that an absolutely continuous function is differentiable a.e.. Is there a function showing that this statement cannot be made into "every AC function is differentiable ...
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27 views

Frechet derivative of exp(X)

I want to show directly from the definition of exp(X) that it is Frechet differentiable at every $X \in \mathbb{R^{n\times n}}$. The definition being exp$(X):= \sum \frac{X^k}{k!}$ I know continuous ...
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39 views

Power Series Approximation of 2-variable Function, compute coefficients

This is my first post here, I'm an undergraduate first year in engineering. We have been given a MATLAB question with a particularly troublesome part that supposes a power series (?) approximation of ...
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0answers
6 views

Backwards Difference Quotient for the Second Derivative?

How is the backwards difference quotient for the second derivative defined and how can I use Taylor to show it is first order accurate? I know that the central difference quotient is definied as $\...
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1answer
38 views

The differentiable equation satisfies the equation…

Struggling with the following, I don't know what to google to get help so I figured, I can ask here. This is translated, so sorry if some parts don't make sense. The differentiable function $y(x)$ ...
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Prove that if $f$ is a linear transformation then $Df(a)(x) = f(x) $

The question and its answer is given below but I do not know why $r = o$, could anyone explain this for me please?
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Use the given data to find an approximation for $f(x)$.

I have the following question in the book: But I do not know how to answer it, the only piece of information that my professor said is that a function differentiable means that it can be approximated ...
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1answer
30 views

Find the derivative of $ \frac{E^2*RL}{(Rs+RL)^2} $

So I have to find the partial derivative $\frac{d}{dRL}$ which means that all variables except RL are constants. In the end I get $\frac {E^2((Rs+RL)^2-2RL)}{(Rs+RL)^3}$ and the answer should be $\...
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0answers
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Why shouldn't we normalize directional derivatives vectors?

Well, here we are. It's my turn. I know this is a heavily disussed topic on this site. Nevertheless, I can't stop myself asking this question. Indeed, my reasearches have left me mostly unsatisfied ...
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2answers
50 views

Question on elementary calculus [duplicate]

Given a function $g(x)$ which has a derivative $g'(x)$ for all $x$ satisfying $g'(0) = 2$ and $$ g(x+y) = e^x g(y) + e^y g(x) $$ for all $x \in \mathbb{R}$. How to show that $$ g'(x) + g(x) - ...
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1answer
44 views

Going from the differential to the derivative (Frechet and matrix calculus)

For a function $f: A \rightarrow B$, Frechet differentiability tells us that we want to find a linear operator that satisfies $$\lim_{H\rightarrow 0} \frac{||f[X+H] - f[X] - G[H]||}{||H||} = 0$$ ...
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At what value of x each of the functions below are Not differentiable? Explain. Function: f(x) = | x -2 | [on hold]

Please try to use derivatives laws in your answer. Otherwise answer freely.
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38 views

Why is that $\int_a^b \frac{\partial f}{\partial x}(x,t)dt = \frac{\partial}{\partial x}\int_a^b f(x,t)dt$

This question is concerned with the integral with parameter, so let's assume that every function below is smooth. To find the formula for the derivative of an integral with parameter, say $$g(x) =...
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4answers
60 views

Proving inequality $x\ln(x)+y\ln(y)\geq(x+y)\ln(\frac{x+y}{2})$

How can we prove that for $x,y>0$ we have $$ x\ln(x)+y\ln(y)\geq(x+y)\ln\left(\frac{x+y}{2}\right) $$ using the derivatives of functions of one variable?
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4answers
45 views

Nth derivative in zero

For $f:\mathbb{R}\rightarrow\mathbb{R}$ where $f(x)=\frac{1}{2+3x^2}$ how can we find out, that $f^{(1001)}(0)=0$?
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2answers
35 views

If $F(x) = f(x)g(x)$, what is th nth derivative of F, that is $F^{n}(x)$, if $f$ and $g$ have derivatives of all orders?

This problem seems really problematic, checking the pattern from the first $5$ derivatives of F, I proceeded to form a general formula, but failed. I did this - $f^{n}(x)g(x) + nf^{n-1}(x)g'(x) + \...