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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Spivak: Show that $f(kx+(1-k)y)<kf(x)+(1-k)f(y)$ whenever $k$ is a rational number, between $0$ and $1$, of the form $\frac{m}{2^n}$.

Here is a starred problem from Spivak's chapter on "Convexity and Concavity" *7. (a) Prove that if $f$ is convex, then $f(\frac{x+y}{2})<\frac{f(x)+f(y)}{2}$. (b) Suppose that $f$ ...
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Gradient of $\mbox{tr} \left (X^T X \right)$

My goal is to compute $$\frac{\mathrm{d} \operatorname{tr}\left(\mathbf{X}^{T} \mathbf{X}\right)}{\mathrm{d} \mathbf{X}}$$ Following the common way of approaching vector/matrix differentiation, I ...
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$f$ convex on interval, prove $f$ is either increasing, decreasing, or there is some $c$ such that $f$ is increas. right of $c$, decreas. left of $c$.

The following is a problem from Spivak's Calculus, in a chapter called "Convexity and Concavity": (a) Suppose that $f$ is differentiable and convex on an interval. Show that either $f$ is ...
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The graphs of $y=ax$ and $y=\arctan(bx)$ intersect at three distinct points if?

I have a question relating to calculus and inverse trigonometric functions. Any help is appreciated. The question is: The graphs of $y=ax$ and $y=\arctan(bx)$ intersect at three distinct points if? A:...
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Algorithmically finding mixed-derivative coefficients using finite differences

Suppose $f : \mathbb{R}^n \rightarrow \mathbb{R}$ is a differentiable (for simplicity, let's assume ${C}^\infty$) function, and I would like to find the following partial derivatives numerically (at ...
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Relationship between Clarke-subdifferential $\partial_{C}f(\, \cdot \,)$ and Bouligand-subdifferential $\partial_{B}f(\, \cdot \,)$

Let $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ be locally Lipschitz sontinuous. (1) The Clarke-subdifferential $\partial_{C} f(x) \subset \mathbb{R}^{n}$ of $f$ in $x \in \mathbb{R}^{n}$ is defined ...
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Prove that if $f$ and $g$ are convex and $f$ is increasing, then $f \circ g$ is convex.

The following is a problem that appears in the appendix to chapter 11 of Spivak's Calculus, entitled "Convexity and Concavity". (a) Prove that if $f$ and $g$ are convex and $f$ is ...
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Unexpected result when calculating shortest distance between point and curve

My math book has following problem: "Find distance between point $(2,0)$ and curve $y=\sqrt{x}$." I assume this asks for the shortest distance because I cannot think of any other distance ...
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Derivative of $g(x) = \frac{\log x}{\log a}$

I am currently reading Spivak's Calculus. I have an older version. In chapter 17 the author presents a definition of the logarithm function. There he presents the derivative of $g(x) = \frac{\log x}{\...
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What is a function such that $f_y(a_1,a_2)=f_{yy}(a_1,a_2)=0$ and $f_{yyy}(a_1,a_2)\neq 0$, and also $f_y(b,a_2)=0$ but $f_{yx}(b,a_2) \neq 0$?

It is easy to construct a function of two variables $f(x,y)$ such that $f_y(a_1,a_2)=f_{yy}(a_1,a_2)=0$ but $f_{yyy}(a_1,a_2)\neq 0$, such as $f(x,y) = -(y-a_2)^3$. It is also easy to construct a ...
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Spotting the existence of asymptotes using calculus?

Are there any requirements that when they are fulfilled (as far as the derivatives of a function are concerned) then a function will always have an asymptote?(a straight line asymptote)(for example ...
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Derivative of $\arcsin(x)=\frac{1}{\cos(x)}$ .

My intuition says that since $\sin(x)$ and $\arcsin(x)$ are inverse of each other, their derivatives must be reciprocal. I have previously proved the fact that $\arcsin(x)'=\frac{1}{\sqrt{1-x^2}}$ but ...
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What is the second derivative of a function on a manifold.

I am a little confused about what the second derivative of a map is. I would like to describe how I understand it and see if I am correct. Let $N,M$ be manifolds and $f:N\to M$ a $C^\infty$ map (for ...
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How to show that $f(x)=\frac{\arcsin x}x$ is increasing when $x\ge0$?

How to show that $f(x)=\frac{\arcsin x}x$ is increasing when $x\ge0$? My Attempt: $f'(x)=\frac{\frac x{\sqrt{1-x^2}}-\arcsin x}{x^2}$ Since $0\le x\le1\implies0\le\sqrt{1-x^2}\le1$ And $0\le\arcsin x\...
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Need to understand what is wrong here in the equality? [duplicate]

Lets's represent this as lhs=rhs x square = x + x + x ... x times. For all positive numbers. derivative(lhs)=derivative(rhs) 2X=X 2=1 This is definitely wrong. Can ...
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What is wrong here in equality? [closed]

Lets's represent this as lhs=rhs x square = x + x + x ... x times. For all positive numbers. derivative(lhs)=derivative(rhs) 2X=X 2=1 This is definitely wrong. Can ...
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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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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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1 answer
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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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1 answer
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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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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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2 votes
2 answers
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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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1 answer
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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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-2 votes
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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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2 answers
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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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2 votes
2 answers
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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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3 votes
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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1 vote
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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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