# 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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### 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^...
1 vote
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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 ...
1 vote
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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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### 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 ...
### 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 ...
### 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: ...