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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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Calculating the derivative of $\sin^3(312x^2)$

Can you explain in detail please how to find the derivative of this function? $$\sin^3(312x^2)$$
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0answers
14 views

$f(x)$ is differentiable or not on the interval?

The question is to use the graph of $f$ to find is f differentiable or not on the given interval? $f(x)=\sqrt{4-x}$ following are the intervals $[0,4]$ and $[-5,0]$ and I have done this question as ...
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0answers
23 views

Multiplying Integrals with Different Bounds

The question I have is two integrals mutliplied with different bounds $$\alpha_0(x)=\int_x^1\exp\left(\frac{-\varphi\lambda^2}2\right)d\lambda\cdot\left(\int_0^1\exp\left(\frac{-\varphi\lambda^2}2\...
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4answers
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Do not understand L'Hopital Rule

I have just started learning L'Hopital rule, and so far I thought I understood everything until I stumbled upon this question $$\lim_{x\to 0} \frac{\ln(\cos(ax))}{\ln(\cos(bx))}.$$ To this, ...
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1answer
20 views

Example of function on R with given property

I wanted to find example of function with following property: $f(x)=0,\forall x\leq 0$ and $f(x)>0,\forall x>0$ I tried but. I stuck in infinite differentiablilty. I thought some combination of ...
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2answers
51 views

A differentiation/derivative/calculus problem

The question is as follows: $$y=x^2/(x+1)$$ The normal to this curve at $x=1$ meets the $x$-axis at point $M$. The tangent to the curve at $x=-2$ meets the $y$-axis at point $N$. Find the area ...
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1answer
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Proof that two definitions of a point of inflection are equivalent

I have seen that many online sources (including other math stack exchange questions) say that the following are equivalent definitions of a point of inflection: If $f$ is differentiable on $I$ we say ...
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2answers
38 views

derivative for parabola

I've been revisiting my calculus and have a simple question I can't seem to answer with respect to derivatives of a parabolic function. Take: $y=x^2$ Derivative $dy = 2x~dx$ However by simply ...
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1answer
78 views

$\dfrac{\partial^2 f}{\partial x \partial y} = 0 \nRightarrow f(x,y) = g(x) + h(y)$

I am working through Ted Shifrin's book Multivariable Mathematics. There is an exercise problem that is meant to demonstrate that one can have $\dfrac{\partial^2 f}{\partial x \partial y} = 0$ but$ f(...
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3answers
42 views

Derivatives of functions composition: $\lim_{x\rightarrow 8}\frac{\root{3}\of{x} - 2}{\root{3}\of{3x+3}-3}$

I have to calculate the folowing: $$\lim_{x\rightarrow 8}\frac{\root{3}\of{x} - 2}{\root{3}\of{3x+3}-3}$$ I am not allowed to used anything else than the definition of the derivative of a function $...
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0answers
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Product of a special $C^\infty (\Bbb R^d)$ and $C^2 (\overline{\Bbb R^d \setminus B(0,R)})$ is $C^2 (\Bbb R^d)$

Let $r > 0$. Suppose $f\in C^2 (\overline{\Bbb R^d \setminus B(0,r)})$, where $B(0,r) := \{ x\in \Bbb R^d : \Vert x \Vert \leq r \}$. This means $f\in C^2 (\Bbb R^d \setminus B(0,r))$ and has a ...
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0answers
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When the derivative is equal to 1?

Looking at reciprocal functions, they have 2 “turning” points, where the higher derivative values near vertical asymptotes turn into the generally lower ones approaching the end behavior asymptote, ...
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3answers
45 views

How to take the derivative with respect to a function? [on hold]

If $f(x)$ is a function, how do I take the derivative with respect to $g(x)$? What I mean is, if say $f(x) = s$, and I want to take the derivative with respect to $g(x) = \log s$, how would I do ...
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0answers
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How to get this first order approximation?

Consider the formula where $Z_t$ and $X_{t+1}$ are vectors. Here, $X_{t+1} = Z_{t+1} - Z_t$. How do we get from this to the following "first-order approximation"? I'm thinkin Taylor's formula, but ...
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1answer
35 views

Prove $\frac{d}{d\theta} R(\mathbf{n}, \theta) = S(\hat{\mathbf{n}})R(\mathbf{n}, \theta)$

I think I am close with this proof - I just need the final step. Let $R(\mathbf{n}, \theta) \in SO(3) $ be the 3D rotation matrix of angle $\theta$ around axis $\mathbf{n}$. Then show that $$\frac{d}{...
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1answer
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Taylor series expansion for $\cos(2x)$ about $\frac{\pi}{8}$

So, knowing that $$f(x) = \sum_{n=0}^\infty \frac{f^n(a)(x-a)^n}{n!}$$ For my case I write $$\cos(2x) = \sum_{n=0}^\infty \frac{\frac{d^n(cos(\frac{\pi}{4}))}{d(\frac{\pi}{8})^n}(x-\frac{\pi}{8})^n}{...
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1answer
30 views

Derivation $f(tx) = t^{\mu}f(x) \text{ } \forall x \in \mathbb{R}^n \text{\ {0}} \text{ } \forall t \in \mathbb{R}^+$

Let $\mu \in \mathbb{R}$ be a real number and $f:\mathbb{R}^n$\ {$0$} $\to \mathbb{R}$ a function that is positive homegenous with degree $\mu$, which means: $$f(tx) = t^{\mu}f(x) \text{ } \forall x ...
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2answers
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Proof: All directional derivatives $\frac{\partial f}{\partial e}$ of $\frac{sin(x^3+y^3)}{x^2+y^2}$ are in the origin

Let $M := (0,\infty) \subset \mathbb{R^2}$ and $f:\mathbb{R}^2 \to \mathbb{R}$. How can one prove that all directional derivatives $\frac{\partial f}{\partial e}$ of $f(x,y)$ are existing in the ...
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1answer
20 views

Proof $\left\lVert g(x) \right\rVert_2 \leq \ln(1+\left\lVert x \right\rVert_2^2) \text{ for all x}\in \mathbb{R}^n $ differentiable

Let $g:\mathbb{R}^n \to \mathbb{R}^m$ with $$\left\lVert g(x) \right\rVert_2 \leq \ln(1+\left\lVert x \right\rVert_2^2) \text{ for all x}\in \mathbb{R}^n $$ How can I prove that $g$ is totally ...
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1answer
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Proving $q:\mathbb{R}^n \to \mathbb{R} \text{ with } q(x):= x^TAx$ totally differentiable

Let $A \in \mathbb{R^n}$ be a real $n \times n$ matrix. How can I prove that the function $$q:\mathbb{R}^n \to \mathbb{R} \text{ with } q(x):= x^TAx$$ is totally differentiable on $R^n$ and find ...
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2answers
21 views

Derivative Function Exercise

If $f(x)=g(x)\cdot(x-1)^2$ then I need to find a $g(x)$ such that there will not be a second derivative of $f$ for $x=1$. I don't know what to search for.
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1answer
22 views

chain rule of a second derivative

Suppose I have the following function where $$z=\omega(\zeta)=\frac{1}{\zeta}$$ and also, $$\phi(\zeta) = \zeta^{-1}+2\zeta$$ By using chain rule, I can get the first-derivative of $\phi(z)$. Notice ...
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3answers
37 views

differentiate $\frac {dx}{dt}$ when $x=(1+e^{-2u})^{-0.5}$

I am stuck in what would be simple differentiation: When $\dot x =\frac {dx}{dt}$ now when $x=(1+e^{-2u})^{-0.5}$ I think I can write: $$\dot x =\frac {d(1+e^{-2u})^{-0.5} }{dt}$$ But can I ...
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1answer
45 views

What's the difference between the two?

Question Which of the following two can define the derivative $f'(a)$: 1)$$\lim_{n \to \infty}n \left[f\left(a+\frac{1}{n}\right)-f(a)\right],n \in \mathbb{Z}.$$ 2)$$\lim_{x \to \infty}x\left[f\...
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0answers
34 views

For which $\alpha \geq 1$ is $\left\lVert x \right\rVert^\alpha$ differentiable?

Let $\left\lVert \cdot \right\rVert_\infty$ be a norm on $\mathbb{R}^n$. How can one find out, for which $\alpha \geq 1$ the image $f$ with $f(x) := \left\lVert x \right\rVert^\alpha$ is totally ...
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2answers
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Obtaining the gradient of a vector-valued function

I have read that obtaining the gradient of a vector-valued function $f:\mathbb{R}^n \to \mathbb{R}^m$ is the same as obtaining the Jacobian of this function. Nevertheless, this function has only one ...
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2answers
70 views

Problem with derivative of $x^{x^x}$

I was recently watching blackpenredpen’s video (found here: https://m.youtube.com/watch?v=UJ3Ahpcvmf8) where he found the derivative of the the function $y = x^{x^x}$. Before watching the video, I ...
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1answer
22 views

Inner Product, Definite Integral

Does the map $<f, g>$ $=$ $\int _0^1\:\left(\left(f\left(x\right)-\frac{d}{dx}f\left(x\right)\right)\left(g\left(x\right)-\frac{d}{dx}g\left(x\right)\right)\right)dx$ define an inner product on ...
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0answers
30 views

Lipschitzness of derivatives

Let $f \in C^\infty_b(\mathbb{R}^d; \mathbb{R}^d)$, so bounded, infinitely differentiable with bounded derivatives mapping $\mathbb{R}^d$ to $\mathbb{R}^d$. I'll write $|\cdot|$ for the norm on $\...
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1answer
20 views

Partial derivative of the likelihood function respect to $\sigma^2$

I am having problem doing the partial derivative of the likelihood function which is $L(\mu,\sigma^2)=\frac{1}{\sigma\sqrt{2\pi}^n}\times \exp{(-\frac{1}{2\sigma^2}\sum(x_i-\mu)^2)}$ If the first ...
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1answer
22 views

Other Points Where Tangent Line Intersects Graph

Q: For each a ∈ $R$ find any other points at which the tangent line ($y = 3a^2 -48$) intersects the original graph ($x^3 - 48x + 2$). Hint: $f(x) − f(a)$ is divisible by $x − a$ Does this just mean ...
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1answer
21 views

Find the derivative of equation in matrix form

I have the following energy function and I need to calculate its derivative with respect to $\alpha$ and I don't know how to do it. I would really appreciate if anyone can help. $$ E=\alpha ^{T}L\...
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1answer
28 views

Proving that cuboid of maximum volume in a sphere is a cube.

I was preparing for my maths test . And preparing application of derivative (theory based question ) there I saw a problem of proving rectangle of maximum area in a circle is square . So there were ...
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3answers
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what does $f '(x^2)$ mean?

enter image description herewhat does $f '(x^2)$ mean? does it mean calculating $f '(x)$ and then putting x=x² or first putting $x=x^2$ in $f(x)$ and then differentiating it? Let $f(x)=x^3$ and $f'(x)...
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2answers
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Continuity and Differentiability of Step function?

All Differentiable functions must be continous , But step function is differentiable and its derrivative is Dirac delta function, Step function actually is not continous But it have Derrivative , How ...
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0answers
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Mathematical topic of functions derivative with respect to own derivative for finding inverse

Is there any topic or branch within mathematics that uses a functions derivative with respect to a different derivative of the same function to solve inverses of functions? E.g. with some function f(...
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1answer
14 views

Find the equation to the tangent of a line using known points?

I have carried out the implicit differentiation of the original formula ($x-y^3=2xy$) to get the equation $$\frac{dy}{dx} = - \frac{2y-1}{3y^2-2x}.$$ Now I need to find the equation of the tangent ...
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2answers
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Optimization Problem. Find Smallest Perimeter Given Area.

QUESTION Find the dimensions of a rectangle with area $1000$m$^2$ whose perimeter is as small as possible. MY WORK I think we are solving for $\frac{dy}{dx}$: \begin{align*} P &= (2x+2y) \\ ...
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2answers
27 views

Directional derivatives of collinear vectors

I'm currently learning about directional derivative, and I need to figure out something in order to fully understand it: What I understood is that directional derivatives are about infinitesimal ...
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3answers
57 views

Showing differentiability of $g(x)=\begin{cases}\frac{f(x)}{x},&\text{$x\neq0$}\\f'(0),&x=0\end{cases}$ given that $f(0)=0$

Let $f$ be a twice-differentiable function of $\mathbb R$ with $f(0)=0$. Define $$g(x)=\begin{cases}\frac{f(x)}{x},&\text{$x\neq0$}\\f'(0),&x=0\end{cases}$$ Prove that $g$ is a differentiable ...
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1answer
27 views

Tangents parallel to one another and finding the perpendicular of a vector function

We have $$x(t)= \begin{pmatrix} 1+t \\ t^2-t \\ 1-t^2 \\ \end{pmatrix}$$ Part 1 Are there $2$ points $x(t_1)$, $x(t_2)$, such that the function’s tangent vectors at these points are parallel ...
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Applying Chain Rule to Dimensionless Transformation

Hello I am trying to show that the equation $\frac{dN}{dt} = rN(1 - \frac{N(t - \tau)}{K})$ can be rewritten in a dimensionless form as $\frac{dy}{dx} = \lambda y(1 - y(x - 1))$ using the ...
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0answers
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Question on differential coefficient as a rate measurer

The question is as follows: O is a given point and NP a given straight line upon which ON is the perpendicular. The radius OP rotates about O with constant angular velocity ω. Show that NP increases ...
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1answer
33 views

Find derivative of a complicated fraction

Let $n> 1$ be an integer and let $x_1,\ldots,x_n$ be positive real numbers, all between $0$ and $1$. Is it possible to find the derivative of it so I know if it is increasing wrt $x_i$? $$ \frac{\...
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0answers
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Geometric interpretation of differential results

I had a question which reads as follows. A triangle has two of its angular points at $(a,0)$ and $(b,0)$, and the third point $(x,y)$ is movable along $y=x$. If $A$ is an area of the triangle; show ...
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1answer
22 views

Basic Implicit Differentiation Problem

QUESTION: Find $\frac{dy}{dx}$ by implicit differentiation. $x^2-4xy+y^2=4$ MY SOLUTION $\frac{d}{dx}x^2 - \frac{d}{dx}4xy + \frac{d}{dx}y^2 = \frac{d}{dx}4$ $2x-(4x)'(y)+(4x)(y)'+2y\frac{dy}{dx} =...
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1answer
19 views

Finding the Shape of a Graph Using the Min and Max

QUESTION: Let $f(x)=2x^{2}-2x^{4}$. Find the open intervals on which $f$ is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima). $f$ is increasing on the ...
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1answer
26 views

Check if multi-variable function is differential

The function is:$$f(x,y,z) = \begin{cases} x^2y^2z^2\sin(\frac{1}{xyz}), & \text{if $xyz\ne0$} \\ 0, & \text{if $xyz = 0$} \end{cases}$$ I need to say where is it differentiable, find the ...
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0answers
12 views

Calculating derivative of any unknown function $f(x)$ at a fixed $x$ using finite difference method

Let's say I have a one variable function $f(x)$ and know its values at some given values of $x$ i.e. I have the following data x=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10] f(x)=[a, b, c, d, e, f, g, h, i,...
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2answers
28 views

Finding analytic expression for $y'(x)$ from $y''(x) = \exp(a\cdot y(x))$

Is it possible to integrate $$y''(x) = \exp(a y(x)) \ \ \ \ \tag{1}$$ to get $$ y'(x) = \frac{\exp(ay(x))}{ay'(x)} + C\ ? \ \ \ \ \tag{2}$$ And then solve the equation $$ y'(x)^2-Cy'(x) = \frac{\exp(...