Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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33 views

At 2:00pm a car's speedometer reads 30mph, and at 2:10pm it reads 35mph. Use the Mean Value Theorem to find an acceleration the car must achieve.

I'm only assuming that f(a) and f(b) are assigned to each respective velocity, but I'm not sure how the mean value theorem can be applied to distance rate and time.
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3answers
57 views

How do I find the derivative of $a^x$?

The homework question I have is find the derivative of $8^{log_2(x)}$ but we haven't learn't how to find the derivative of a function where the variable is the power of a constant. How do I do this?
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0answers
38 views

Derivative of the Gauss map is zero

If the derivative of the Gauss map is zero in every point in the image of a given local chart, can I conclude that the normal vector is constant and such image is contained in a plane? Edit: The ...
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2answers
29 views

Straight vs Partial derivative

Does it make sense to write $\frac{d}{dx}u(x,t)$ or can one only write $\frac{\partial}{\partial x}u(x,t)$?
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1answer
29 views

Finding concavity using a second derivative that is never greater than zero.

When is $\frac{3x-8}{4(2-x)^{3/2}} > 0$? The equation above is the second derivative of the function: $$f(x) = x\sqrt{2-x}$$ I am wanting to find the concavity of the original function. I know ...
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2answers
31 views

Using the binomial theorem to generate a geometric proof of the derivative.

According to wikipedia, if we wanted to prove $$(x^n)'=nx^{n-1}$$ geometrically by creating an $n$-dimensional hypercube $$(x+\Delta x)^n$$ and setting $a=x$ and $b=\Delta x$, we could expand using ...
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1answer
29 views

Algebraic vs. analytic definition of the multiplicity of a polynomial's root

Let $f(x) = a(x - c_1)^{d_1}(x - c_2)^{d_2} \dots (x - c_n)^{d_n}$ be a polynomial over the complex numbers ($n, d_i \in \{1, 2, \dots\}$, $a \in \mathbb{C}\setminus \{0\}$), where the roots $c_1, ...
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3answers
55 views

Find the derivative of $2^x-3^y=1$ and then find the limit of $dy/dx$ as $x\to\infty$

Please tell the answer If $2^x-3^y=1$ then what is the value of $$\lim_{x\to\infty} \frac {dy}{dx}?$$ I have tried finding the derivative implicitly, but I only get $0$ on both sides.
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2answers
35 views

Vertical asymptotes of a given non-rational radical funtion

We have that $f$ is a function $f(x) = x\sqrt{x+4}$. Hence, $f'(x) = \dfrac{3x+8}{2\sqrt{x+4}}$. Then, $\lim_{x \to -4^+}f'(x) = -\infty$. This means that $f$ has a vertical slope at $f(-4)$. It ...
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2answers
87 views

Derivative of a function with respect to a matrix

I need to calculate derivative of the following function with respect to the matrix X: $f(X)=||diag(X^TX)||_2^2$ where $diag()$ returns diagonal elements of a matrix into a vector. How can I ...
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3answers
100 views

Derivative of sin(x)/x at $0$ by definition of derivative

the question I am attempting is: Show $f '(0) = 0$ for: $$f(x) = \left\{ \begin{array}{lr} \frac{\sin(x)}{x} & : x \neq 0\\ 1 & : x=0 \end{array} \right.$$ So I got stuck after the ...
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1answer
88 views

How can I prove that no solution (except y=0) cannot be extended for all t axis?

Given this ODE: $$\dot{y} = (2 + \cos y(t)) \cdot{y^2(t)}$$ I need to show that no solution of this equation holds for every $t \in \mathbb{R}$, and I don't even know how to start. Any hints will do.
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2answers
345 views

A differentiation question conceptual query

I'm quite unsure about how to deal with differentiation of absolute functions, and their continuity. For example, the question I was dealing with was the following: $$ f(x) = \frac{x}{1 + |x|}$$ ...
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1answer
22 views

Related Rates Question Concerning Boyle's Law

Boyle's Law for enclosed gases states that if the volume is kept constant, the pressure P and temperature T are related by the equation P/T=k, where k is a constant. Suppose that the rate of change of ...
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2answers
44 views

If $(g'(x))^2 = g(x)$…

The question is if $(g'(x))^2 = g(x)$ for all real $x$ and $g(0) = 0$, $g(4) = 4$, then $g(1)$ equals... The answer is 1/4. I was thinking if you plugged in the numbers, then that would mean $g'(0) ...
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1answer
30 views

Why $f_{xy}(x_c,y_c)=f_{yx}(x_c,y_c)$?

Why $f_{xy}(x_c,y_c)=f_{yx}(x_c,y_c)$? I found this used here (in the definition of B). $x_c,y_c$ are critical points.
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2answers
39 views

Finding Equation of tangent line

Can someone double check my work to see if I'm doing it correctly? Find the equation of the line tangent to the graph of $(2,1)$ where $f$ is given by $f(x) = 2x^3 - 2x^2 + 1$ 1) $f'(x) = 6x^2-4x$ ...
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2answers
38 views

how to find tangent line at a given point, without equation

Find the equation of the line that is tangent to the curve at the point $(0,\sqrt{\frac{\pi}{2}})$. Given your answer in slope-intercept form. I don't know how can I get the tangent line, without a ...
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1answer
17 views

Using Taylor's Theorem to expect maximum error

Suppose that we use $p(x) = 1 + x + \frac {x^2} 2$ as an approximation for $f(x) = e^x$ on the interval (-.5, .5). What's the maximum error we can expect. I know that Taylor's Theorem states Let ...
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2answers
53 views

Calculus Finding the derivative

I have a practice problem that says: Let $$j(x) = \frac{g(x)}{f(x)}$$ Find $j'(1)$: I don't know how to do this. The answer the book has is $-2$. What I tried to do was set $j(0) = ...
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1answer
48 views

Complex function of class $C^m$

Let $m$ a positive integer and consider the function $$f(z)=\vert z\vert^\alpha z$$ with $\alpha>0$. I have to find the value of $\alpha$ for which $f\in C^m(\mathbb{C},\mathbb{C})$. Now if ...
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2answers
49 views

How do I see that a linear function $L: \mathbb R^n \rightarrow \mathbb R^m$ is smooth?

How do I see that a linear function $L: \mathbb R^n \rightarrow \mathbb R^m$ is smooth ? I see that $L$ is indeed differentiable with $(DL)_x = L$ by definition of the derivative. But how do I ...
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1answer
68 views

Differential identity $\left(x^2\frac{d}{dx}\right)^nf(x)=x^{n+1}\frac{d^n}{dx^n}\left(x^{n-1}f(x)\right)$

I have found the following differential identity: $$\left(-x^2\frac{d}{dx}\right)^nf(x)=(-1)^n x^{n+1}\frac{d^n}{dx^n}\left(x^{n-1}f(x)\right)$$ I have used it to find an alternative Rodrigues ...
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3answers
84 views

Proving that the second derivative of a convex function is nonnegative

My task is as follows: Let $f:\mathbb{R}\to\mathbb{R}$ be a twice-differentiable function, and let $f$'s second derivative be continuous. Let $f$ be convex with the following definition of ...
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3answers
3k views

Why, although these functions have the same derivative, do they not differ by a constant?

I calculated the derivative of $\arctan\left(\frac{1+x}{1-x}\right)$ to be $\frac{1}{1+x^2}$. This is the same as $(\arctan)'$. Why is there no $c$ that satisfies $\arctan\left(\frac{1+x}{1-x}\right) ...
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0answers
55 views

Prove there exists a infinitely differentiable function whose value of partial derivatives of all orders at $0$ is a given function

Let $n \geq 1$ be an integer, and $C: \mathbb{N}^n \to \mathbb{R}$ be a function. Prove that there exits a infinitely differentiable function $f: \mathbb{R}^n \to \mathbb{R}$ whose value and partial ...
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4answers
408 views

Could such a polynomial/rational function exist? [closed]

Is it possible for there to be a function, which has "nice values" for its $x$-intercepts, local extremes, and inflection points? By nice values, I essentially mean that all the $x$-intercepts, local ...
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2answers
35 views

Parametric differentiation

The parametric equations of a curve are $$\begin{cases}x(t)=e^{-t}\cos t\\y(t)=e^{-t}\sin t\end{cases}$$ Show that $$\frac{dy}{dx}= \tan\left(t-\frac{\pi}{4}\right)$$ I did the differentiation ...
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0answers
31 views

Taylor expansion of a power function

I was wondering about Taylor expansions of functions of the form $x^p$, where p is a real number, about $x = 0$. It seems clear how to do it about any other point, but what happens to the series as I ...
2
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3answers
44 views

differentiation of a matrix function

In statistics, the residual sum of squares is given by the formula $$ \operatorname{RSS}(\beta) = (\mathbf{y} - \mathbf{X}\beta)^T(\mathbf{y} - \mathbf{X}\beta)$$ I know differentiation of scalar ...
2
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2answers
129 views

Determining the rate of change of a radius as a sphere loses volume

Problem: A spherical balloon leaks $0.2\mathrm m^3 / \mathrm{min}$. How fast does the radius of the balloon decrease the moment the radius is $0.5\mathrm m$? My progress: Since we're dealing with ...
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6answers
132 views

How to differentiate $y=\sqrt{\frac{1+x}{1-x}}$?

I'm trying to solve this problem but I think I'm missing something. Here's what I've done so far: $$g(x) = \frac{1+x}{1-x}$$ $$u = 1+x$$ $$u' = 1$$ $$v = 1-x$$ $$v' = -1$$ $$g'(x) = \frac{(1-x) ...
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0answers
29 views

Verifying proof that $f$ has an inflection point at zero if $f$ is a function that satisfies a given set of hypotheses.

Prove that if $f$ is a function $f(x): f'(x) > 0$ $\forall x: x \neq 0 \wedge f'(0) = 0 \wedge f''$ is a continuous one to one function on some open interval $(a, b): a < 0 < b$ then ...
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2answers
77 views

How to find the derivative of an integral where both, the limit and the integrand, are functions of x?

I found a good expository paper by Keith Conrad, which explains by examples the technique of derivative under the integral sign. Here's the link: ...
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1answer
32 views

solving differential equation with delta fucntion wihout transformation

How Can I solve this problem without Laplace transform.... My professor said that I can s olve the differential equation problem with delta fucntion forcing by intergrating over the momentum of ...
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3answers
79 views

How to evaluate this exponential fraction limit?

I am trying to determine if 3$^n$ grows faster than 2$^{2n}$. One way I found online to do this was, from Growth was to evaluate $\lim_{n\to \infty} \frac{3^n}{2^{2n}}$ and if that limit evaluates ...
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2answers
36 views

Does dimension of derivative should agree with dimension of variable?

Suppose variable $x$ is a N*1 vector, $A$ is a M*N matrix and $b$ is a M*1 vector. $$ f(x) = \|e^{Ax} - b \|_2^2 $$ Does its derivative should be like following? $$ \frac{\partial f}{\partial x} = ...
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1answer
45 views

Notation for a derivative

I am interested if there is notation for a derivative that is in between a total derivative and partial derivative. The total derivative of $f(t,x,y)$ with respect to $t$ is $$ ...
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1answer
86 views

If $f+f'<\varepsilon$, then $f'<\varepsilon$

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$ and there exstis $\varepsilon>0$ such that $f(x)+f'(x)<\varepsilon$ for all $x\in (a,b)$. Prove that $f'(x)<\varepsilon$ for ...
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1answer
31 views

Looking for a standard function $y(x)$ when $dx/dy = ay+b$

I am looking for a standard function $y(x)$, for which the following applies: $dx/dy = ay + b$, with $a$ and $b$ both being constants. Many thanks in advance!!!
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0answers
27 views

Proof that derivative operator cannot be written in terms of composition operator (without limits)

Difference delta operator can be written without a limit: $$\Delta[f(x)]=f(x+1)-f(x)$$ The same is true for any other finite difference operator. But what about derivative? Is there a proof that it ...
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1answer
43 views

derivatives of non-analytic smooth functions

I would like to know how to calculate the derivative of a non-analytic smooth function? Suppose $f:\mathbb R\rightarrow \mathbb R$ is in $\mathcal C^\infty\backslash \mathcal C^\omega$ and in ...
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0answers
34 views

acoustic dipole volume integral w/ dirac delta?

I have an acoustic research problem that leads to the following integral formulation: \begin{align} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}p(\mathbf{y},\tau)\frac{\partial}{\partial ...
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1answer
46 views

Value of x of which a slope is undefined for a parametric graph.

For what values of $x$ is the slope undefined for the graph $$x=8-t^3$$ $$y=t^2-6t$$ The slope should be undefined when $\frac {dx}{dt}=0$. $$\frac {dx}{dt}=-3 t^2$$ $$-3t^2=0$$ $$t=0$$ When ...
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1answer
80 views

Continuity of left derivative implies differentiability?

Suppose $f:\mathbb{R}\rightarrow \mathbb{R}$ is continuous and has a left derivative, $f^-$, everywhere in a neighborhood of $x.$ Suppose $f^-$ is continuous at $x.$ Does this imply that $f$ is ...
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0answers
16 views

Is $\int_{0}^{1}\nabla f({t\mathbf{x}})\mathrm{d}t$ ($t\in\mathbb{R}$, $\mathbf{x}\in\mathbb{R}^n$) differentiable with respect to $\mathrm{x}$?

Consider a differentiable vector function $f:\mathbb{R}^n\longrightarrow\mathbb{R}$. Is the single-variable integral of its gradient $$\int_{0}^{1}\nabla f({t\,\mathbf{x}})\,\mathrm{d}t\qquad ...
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1answer
23 views

Finding derivatives for a Cauchy-Euler ODE

I'm having some trouble following along with the reduction of the Cauchy-Euler equation into a linear one with constant coefficients. I've been trying to follow along with the work here, but I don't ...
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1answer
44 views

Trick with differentials from $\frac{dr}{ds} \to \frac{dr}{dt}$

I need to come from $r''=\frac{d^2r}{ds^2}$ to $\ddot{r}=\frac{d^2r}{dt^2}.$ I know that $r' = \frac{\dot{r}}{|\dot{r}|}.$ And more $1/\dot{s} = t' = \frac{1}{\sqrt{\dot{r}\cdot\dot{r}}}.$ You can see ...
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2answers
43 views

What are the critical points of $x-4\sqrt{x+1}$?

A critical point $c$ is defined as $f'(c) = 0$ or $f'(c) = $ undefined. This definition is taken from this video. if $$f(x) = x-4\sqrt{x+1}$$ then $$f'(x) = 1 - \frac{2}{\sqrt{x+1}}$$ To find the ...
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2answers
74 views

Prove that $\nabla_X tr(X^TAX)= (A+A^T)X $ where $A \in \mathbb{C^{m*m}}$ and $X \in \mathbb{R^{m*n}}$

Prove that $\nabla_X tr(X^TAX)= (A+A^T)X $ where $A \in \mathbb{C^{m*m}}$ and $X \in \mathbb{R^{m*n}} $ . 1.) Same proof stands when $ A\in \mathbb{C}$ or $ A\in \mathbb{R}$ ? 2.) What is the ...