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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In what framework is it okay to swap the derivative in a product within an integral? [Viscoelasticity]

Dear people with an affinity for math, I am just an engineer approaching the field of viscoelasticity. Currently, I would like to understand the derivation of the generalized Kelvin-Voigt material. It ...
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Knowing $f(t^\ast)\ge 0$ (and some other information), can we show that $f(t)\ge 0$ at $t<t^\ast$?

I asked a similar question before and had to make several changes so before anyone spends time on answering it, I decided to clarify here. We have constants $\alpha_1,\alpha_2,r_1,r_2,c>0$ and we ...
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Evaluating the value of |$f'(x)$| [duplicate]

Let $f \in C^2[0,2]$ and $|f(x)| \le 1, |f''(x)|\le1$ for all $x \in[0,1]$ Prove that $|f'(x)|\le2 $ for all $x \in [0,2]$ I tried Taylor expansion to evaluate $|f'(x)|$, but it seems to work only ...
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Maclaurin series, find the tenth derivative

The problem is as follows: Find the Maclaurin series of $$\begin{cases} \frac{\sin(x)}{x},& x \neq 0 \\ 1,& x=0 \end{cases}$$ and then find $f^{10}(0)$. I figured out the series, if $x\neq 0$ ...
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how do i solve non linear equation? TOPIC IS PDE [closed]

Find a separated solution of the following nonlinear wave equation: ∂u/∂t=cu ∂y/∂x and What is a separated solution of the 2 -dimensional wave equation (∂^2 u)/(∂t^2 )=a (∂^2 u)/(∂x^2 )+b (∂^2 u)/(∂y^...
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Deriving in variational calculus

In this article$^\color{red}{\star}$, page 5: equation (3.10) second line of the equation $$\mathcal{L}= \dots -\mu w \bigg[\dots + \int_0^Dx'ds\times\int_0^Dy'ds - \dots \bigg]\dots$$ then, after ...
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Boundedness of the gradient of Cross Entropy Loss

When analyzing the convergence of algorithms, the assumption of the bounded gradient is often used. I wondered if this holds in the case of cross-entropy loss; otherwise, is there a way to ensure that ...
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Extrapolate impression values from total impressions.

Im looking to extrapolate impression values for specific page level URLs however I only have the total impressions for a domain, and the ranking of the URLs (where the closer you are to 1 the higher ...
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How to express a function into powers of $(x-1)$ and $(y-2)$ using Taylor's formula?

Use Taylor's formula to express the following in powers of $(x-1)$ and $(y-2)$: $f(x,y)=x^3 + y^3 + xy^2$ Solution: $f(1,2)=1 +8 + 4=13$ $f_x (1,2) = 3 + 4=7$ $f_y (1,2) = 12 + 4=16$ $f_{xx} (1,2) = 6$...
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how to derive the following func'

Suppose that the functions $f(x) , g(y) ,h(y)$ are continuous and derivative. How to derive the following function? $$\frac{d}{dy}\int^{g(y)}_{h(y)}f(x)dx$$
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For $a<x<b$, given definition of convexity as $\frac{f(x)-f(a)}{x-a}<\frac{f(b)-f(a)}{b-a}$, is $\frac{f(b)-f(x)}{b-x}>\frac{f(b)-f(a)}{b-a}$ true?

Given the following definition of a convex function $f$ A function $f$ is convex on an interval if for $a,x$, and $b$ in the interval with $a<x<b$ we have $$\frac{f(x)-f(a)}{x-a}<\frac{f(b)-...
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Concavity of a Derivative from a Graph

Is it possible to know the concavity of a derivative $f'$ given the graph of $f$? For example, I was given this graph here: How can I know the concavity of the function $f'$ over $(0,1)$ by simply ...
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Spivak's Calculus, Ch. 11, **69b: $f$ increasing at every $a \in [0,1]$. Prove $f$ increasing on $[0,1]$.

A function $f$ is increasing at $a$ if there is some number $\delta>0$ such that $$f(x)>f(a) \text{ if } a<x<a+\delta$$ and $$f(x)<f(a) \text{ if } a-\delta<x<a$$ (a) Suppose ...
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Confusion on how Covariant Derivatives work on Vector fields

I'm currently watching these Lectures on General Relativity and I tried to work out a simple example to help me understand the content of the lectures better. I tried to calculate the acceleration of ...
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Doubt regarding finding the gradient of of a scalar field

I am new to vector calculus. I watched few you tube videos and came to the conclusion that directional derivative is something like slope with direction and its ...
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Spivak's Calculus, Ch. 11, **69a: $f$ continuous and increasing at every $a \in [0,1]$. Prove $f$ increasing on $[0,1]$.

**69. A function $f$ is increasing at $a$ if there is some number $\delta>0$ such that $$f(x)>f(a) \text{ if } a<x<a+\delta$$ and $$f(x)<f(a) \text{ if } a-\delta<x<a$$ (a) ...
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Why are the product rule and quotient rule for differentiation distinct concepts.

I am undergraduate student studying physics, but interested in math. I am working through a derivation of electric quadrupolarization. In this derivation, I need to differentiate a ratio of functions. ...
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How to compute the derivative of this matrix equation

The matrix $\mathbf{A}(c)$ with the dimension $M \times N$, $c$ is a scalar variable. The matrix $\mathbf{d}$ is a constant matrix with the dimension $M \times 1$. If the formula $\frac{d\mathbf{A}(c)}...
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Spivak's Calculus, Ch. 11, **68: $f(x)=\alpha x+x^2\sin{1/x}$ for $x \neq 0$, $f(0)=0$. Prove $f$ is not increasing in an interval around $0$.

Two asterisks on a problem in Spivak's Calculus signal a potentially very tricky problem. I solved the following two asterisk problem from chapter 11, "Significance of the Derivative". I am ...
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What is an expansion for the $k$th order derivative of a dot product?

Suppose I have two smooth curves $\vec{x}(t)$ and $\vec{y}(t)$ of equal finite dimension. Using the product rule, their first derivative is shown in this post to be $$\frac{d}{dt}\Bigl( \vec{x}(t) \...
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Formula for $k$th order derivative of the determinant of a matrix?

Background Jacobi's formula tells us that $$\frac{d}{dt}\det A(t) = \operatorname{tr} \left( \operatorname{adj}(A(t)) \frac{dA(t)}{dt} \right) = (\det A(t)) \cdot \operatorname{tr} \left( A(t)^{-1} \...
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Solving $ A\frac{\partial z}{\partial x} + B \frac{\partial^2 z}{ \partial x \partial y} + C \frac{\partial^3 z}{\partial x \partial^2 y} = 0 $? [closed]

Non-mathematician here trying to find a hopefully analytic solution or any constructive directions for solving differential equations of this particular form: Take a function $z(x,y)$, is there any ...
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What is $f'(\frac{1}{f(x)})$ in Leibniz's notation where $y=f(x)$ and $f(x)$ is differentiated with respect to $x$

I'm trying to find differentiable function whose reciprocal equals its inverse $f^{-1}(x)=[f(x)]^{-1}$. I read that there is no such function, but I still wanted to try. If the equality is true, then $...
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Are there nonlinear differentiable functions that are positively homogeneous of order 1?

A function $f:\mathbb{R}\mapsto \mathbb{R}$ is positively homogeneous of order 1 if $f(tx) = tf(x) \quad \forall t>0$. For instance, $f_{\alpha}(x) = \alpha x$ is a positively homogeneous funnction ...
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How do we rigorously prove that for $n>1$, $(1+x)^{n-1}<1$ for $-1<x<0$?

Given $n>1$ and $$(1+x)^{n-1}<1$$ Intuitively I can see that for $x \in (-1,0)$, we have $1+x<1$, and if we raise that to any power then it will be smaller than 1. How do we prove this ...
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the derivative of $\sqrt[\beta]{x^\alpha}$

Please someone tell me what it's wrong with this procedure $\frac d{dx} \sqrt[\beta]{x^\alpha} = \frac d{dx}x^\frac \alpha\beta$ \begin{align*} \frac d{dx} x^\frac \alpha\beta & = \frac \alpha\...
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Spivak's Calculus: Using derivatives prove that if $n \geq 1$, then $(1+x)^n > 1+nx$ for $-1<x<0$ and $x>0$.

The following is a problem from Spivak's Calculus, Ch. 11 Use derivatives to prove that if $n \geq 1$, then $$(1+x)^n > 1+nx, \text{ for } -1<x<0 \text{ and } x>0$$ (notice that equality ...
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How differentiate signals

I am doing a course called Signals & Transform and I am having difficult time understanding the concept of unit step functions and how to use it to differentiate signals. Here is the exercise: ...
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derivative of determinant, solve singularity equation for a variable

I would appreciate any tipps on the following problem I really struggle with: For $A \in \mathbb{R}^{n_2\times n_1}, B \in \mathbb{R}^{n_3\times n_2} , \lambda \geq 0, L \geq 0.$ I want to find a ...
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$f''(x)$ is always positive then $f(x+f'(x)) \geq f(x) $ [duplicate]

$f: R \to R$ be such that $f''(x) >0$. Prove that $f(x+f'(x)) \geq f(x) $. My thought: $f''(x) >0$ means f is concave up but $f'(x) $ can be either positive or negative or may be mixed (positive ...
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Show that $f(x)=x^2+\frac{a}{x}$ cannot have a local maximum for any value of $a$.

Show that $f(x)=x^2+\frac{a}{x}$ cannot have a local maximum for any value of $a$. I have tried to find the derivative here. $$f(x)=x^2+\frac{a}{x}$$ $$f'(x)=2x-\frac{a}{x^2}$$ As, derivative is zero ...
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1 answer
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Find the derivative of $\frac{d}{dx}\left(\tan \left(\sqrt{x}\right)\right)$ - no chain rule

I'm trying to find the derivative of: $$ \frac{d}{dx}\left(\tan \left(\sqrt{x}\right)\right) $$ As per the chain rule I have to find the derivative of $tan()$ and then $(\sqrt{x})$ which at the end is ...
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Prove that if $y \neq 0$ and $n$ is odd, then $x^n+y^n=(x+y)^n$ only if $x=0$ or $x=-y$ (Spivak's Calculus, Ch. 11)

The following is a problem from Spivak's Calculus, ch. 11, "Significance of the Derivative". (b) Prove that if $y \neq 0$ and $n$ is odd, then $x^n+y^n=(x+y)^n$ only if $x=0$ or $x=-y$. My ...
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Spivak Calculus Ch. 11, Prob. 61a: $f$ diff in interval containing $a$, $f'$ discont. at $a$. Prove one-sided limits of $f'$ at $a$ cannot both exist.

The following is a problem from ch. 11 of Spivak's Calculus Suppose that $f$ is differentiable in some interval containing $a$, but that $f'$ is discontinuous at $a$. Prove the following: (a) The ...
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Suppose $g = f \circ u$ with $u$ and $g$ smooth but $f$ not smooth, does this imply $g' = u' = 0.$

Prove or provide a counterexample. Consider $g = f \circ u$ two equivalent continuous curves on an open interval $\mathrm{I}$ with values in $\mathbf{R}^d$ ($d = 1$ is OK). So $u$ is assumed ...
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5 votes
3 answers
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If $ax^2 + 2hxy + by^2 = 0$ (here $a, b, h$ are real constants), then find $\frac{dy}{dx}$.

Question: If $ax^2 + 2hxy + by^2 = 0$ (Where $a, b, h$ are real constants), then find $\dfrac{dy}{dx}$. Following choices are given:- $\dfrac yx$ $\dfrac xy$ $\dfrac {-y}x$ $\dfrac {-x}y$ My work: ...
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How to find the directional derivative at every point?

Given a function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $$f(x,y) = \begin{cases} \left( x^2 + y^2 \right) \cos \left(\frac{1}{x^2 + y^2} \right) & \text { if } (x,y) \neq (0,0)\\ 0 & \...
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Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$?

here's my question If $y^2=ax^2+bx+c$ then prove that $y^3[d^2y/dx^2]$ is a constant . I have solved this using the conventional method, taking square root, differentiating w.r.t to x and using ...
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$f$ cont. at $a$, $f'$ exists in interval containing $a$ (except possibly at $a$), $l=\lim\limits_{x \to a^+} f'(x)$ exists. Does $f'(a)=l$?

This question regards the following theorem (as stated in Spivak's Calculus): Theorem 7: Suppose $f$ is continuous at $a$, $f'(x)$ exists for all $x$ in some interval containing $a$, except perhaps ...
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1 vote
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The way of writing the second derivatives of $y$ with respect to $x$, in Leibniz notation [duplicate]

The way of writing the second (or higher) derivative of $y$ with respect to $x$, in the Leibniz notation is $\dfrac{d^2y}{dx^2}$. Why $d$ on face takes number $2$ as its power, but in denominator $d$ ...
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3 votes
1 answer
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What is meant by "the image of an interval is also an interval" when this phrase is used to describe the intermediate value property?

Wikipedia says of the Intermediate Value Theorem that In mathematical analysis, the intermediate value theorem states that if $f$ is a continuous function whose domain contains the interval $[a, b]$, ...
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In Darboux's Theorem, no assumption is made that $f'$ is continuous. In $[a,b]$, what is an example of $f$ differentiable, $f'$ not continuous? [duplicate]

What is an example of $f$ differentiable on $[a,b]$ but $f'$ is not continuous at some point in $[a,b]$? For context about why I am asking: Darboux's Theorem says that if $f$ is differentiable on $[a,...
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$f$ is diff. in some interval containing $a$, $f'$ is discontinuous at $a$. Can one-sided limits of form $\lim\limits_{x \to a^+} f'(x)$ both exist?

The following is a problem from ch. 11 of Spivak's Calculus Suppose that $f$ is differentiable in some interval containing $a$, but that $f'$ is discontinuous at $a$. Prove the following: (a) The ...
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If $f(x)=1$ for $x \geq 0$, $f(x)=-1$ for $x < 0$, what is the correct way to draw the graph of $f'$ at $0$?

Consider $f$ and $f'$ as depicted in the picture below Is the correct depiction of $f'$ the second or the third graph? $f(0)$ is $>0$ in the first graph, and if the derivative is defined as a ...
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Is, for all functions $f(x), g(x),\;\lim_{x\to0^{+}} \frac{[x]+f(x)}{g(x)}=\lim_{x\to0^{+}}\frac{f(x)}{g(x)},$ true?

Can we say that for all functions $f(x)$ and $g(x)$ the following holds true: $$\lim_{x\to0^{+}} \frac{[x]+f(x)}{g(x)}=\lim_{x\to0^{+}}\frac{f(x)}{g(x)}$$ where $[x]$ is the greatest integer function. ...
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1 vote
3 answers
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If $f$ differentiable at $a$, $\lim\limits_{x \to a^+} \frac{f(x)-f(a)}{x-a} \geq 0$, what is the exact argument that allows us to say $f'(a) \geq 0$?

The following is a problem from Ch. 11 in Spivak's Calculus: (a) Suppose that $f$ is differentiable on $[a,b]$. Prove that if the minimum of $f$ on $[a,b]$ is at $a$, than $f'(a) \geq 0$, and if it ...
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Undetermined coefficients & numerical differentiation

so I have to obtain finite difference formulas and perform numerical experiments such as: ▶ Input: stencil, requested accuracy, point of interest ▶ Output: finite difference formula and I should ...
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Interpolation that has non-zero second derivatives?

I'm wondering if there are any methods of 2D interpolation that are twice differentiable whose second unmixed partial derivatives are not zero? The use case is to compute the Hessian matrix of an ...
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Solve $x \frac{d^{2}y}{d x^{2}}+\frac{d y}{d x}+x y=0$

Solve the differential equation $$x \frac{d^{2}y}{d x^{2}}+\frac{d y}{d x}+x y=0$$ My try: Let $z := x \frac{dy}{dx}$. So, we get $$\frac{dz}{dx}=x\frac{d^2y}{dx^2}+\frac{dy}{dx}$$ and, thus, $$\frac{...
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Can we solve this differential equation or there is some other way [closed]

$f(x)$ is defined for $x≥0$ and has a continuous derivative. It satisfies $f(0)=1,f'(0)=0$ and $$(1+f(x))f''(x)=1+x$$ Then which of the following is not a possible value of $f(1)$? $2$ $1.75$ $1.5$ $1....
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