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

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0
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2answers
43 views

Antidifferentiation: Stone dropped from $150ft$ rising at $10ft/sec$

A stone is dropped from a balloon when it is $150ft$ above the ground and rising at the rate of $10ft/sec$. How long will it take the stone to strike the ground, and with what velocity does it strike ...
2
votes
4answers
50 views

Antiderrivative of ${d^2 y \over dx^2} = 1-x^2$

At any point $(x,y)$ on a curve, ${d^2 y \over dx^2} = 1-x^2$, and an equation of the tangent line to the curve at the point $(1,1)$ is $y=2-x$. Find an equation of the curve. This is what I've done ...
1
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3answers
42 views

Differentiability of piecewise functions

Check whether the function is differentiable: $$f:\mathbb{R}^2\rightarrow \mathbb{R}$$ $$f= \begin{cases} \frac{x^3-y^3}{x^2+y^2} & (x,y)\neq (0,0) \\ 0 & (x,y) = (0,0) \\ \end{...
0
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0answers
43 views

Calculus: Derivative of a summation and dot product

I'm trying to implement a speed boost to an eye-tracking algorithm (found here: http://www.inb.uni-luebeck.de/publikationen/pdfs/TiBa11b.pdf). I need to take the derivative of the eye-tracking ...
2
votes
1answer
50 views

Neural Network - Why use Derivative

Good Day I am trying to get an understanding of Neural Network. Have gone through few web sites. Came to know the following: 1) One of main objective of neural network is to “predict” based on data....
0
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1answer
43 views

Differential equation without analytic solution - comparative statics

I am facing a differential equation - with boundary condition $v(T)$ given - without an analytic solution but still need to understand how the solution is affected by a change of the function's value. ...
1
vote
1answer
49 views

Concavity of function $\sin(x^2)$.

I want to check where the function is convex and where concave. For this I need to calculate the second derivative test: I got $f''(x) = 2*\cos(x^2) -4*x^2(\sin(x^2))$ and this derivative should be ...
0
votes
1answer
24 views

Chain rule confusion(multivariable functions)

I am confused bz the chain rule of multivariable function. I know, that sometimes it is impossible to dot it. For example i have the following functions: $f(x_1,x_2) = \begin{bmatrix} x_1x_2^2 + x_1^...
2
votes
1answer
100 views

Why is $\frac{d(x^n)}{d(x)}=nx^{n-1}$

So I was thinking about what I have learnt and I realised that I kind of took the derivative of a function for granted. So I did some research as I wanted to find out how this was discovered and I ...
0
votes
2answers
33 views

Confusion about the different ways of writing Taylor Polynomials

For the sake of using a simple example, let's say I want to approximate $y=x^3$ with a second degree polynomial, and let's say I want to construct my polynomial around the point $x=4$. One way I ...
0
votes
3answers
29 views

Derivatives: Combining Product & Chain Rules

So I'm working through the material on Khan Academy, and the question is: "Consider the function $f(x) = x^n\ln x$, defined for $x > 0$. Determine, in terms of $n$, the value of $x$ for which $f'...
1
vote
2answers
49 views

Question about proof: continuity of partial derivatives implies total differentiability

I have a lack of understanding regarding this proof, and since the proof is not in English, I will simply write it down up to the point where I can't go further: Statement: Assume $U \subset \Bbb R^...
1
vote
1answer
30 views

Bounding a $C^0$ function with $C^1$ functions

Given a continuous function from $(0,+\infty)$ in itself, with $\lim_{x\to 0^+} f(x)=0$, find $C^1$ functions $g,h:(0,+\infty)\longrightarrow(0,+\infty)$ such that $g\leq f\leq h$ and $\lim_{x\to 0^+}...
1
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2answers
41 views

Partial Derivatives Approximation

By definition we know the following: \begin{equation} \frac{\partial f(x,y)}{\partial x} \approx \frac {f(x+ \delta x,y)-f(x,y)}{\delta x} \end{equation} \begin{equation} \frac{\partial f(x,y)}{\...
-1
votes
0answers
14 views

Is there any equality for the integral of the product of normal derivative?

I am trying to get the proof of $\int\int_DD_uf(x) D_ug(x) dx$. For example in Green Theorem, in integral we use the product of $ \nabla$, when it comes to normal derivative, how can I organize the ...
2
votes
1answer
87 views

Second Differential

Let $(x,y,z)$ a coordinate system, $M=\mathbb{R}^3$ and we also denote by $x$ the first coordinate function : $x:M \rightarrow \mathbb{R},\; q=(a,b,c) \mapsto a$. We have $dx:TM \rightarrow \mathbb{R}...
0
votes
2answers
94 views

Minimum value of $4a+b$

Let $ax^2+bx+8=0$ be an equation which has no distinct real roots then what is the least value of $4a+b$ where $a,b\in \Bbb R$. My Try: I differentiated the given function to get $f'(x)=2ax+b$ now ...
-1
votes
0answers
57 views

Analytical solution for a Non-linear differential equation $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$

Is there any analytical solution for the following differential equation? $\frac{d^2y}{dt^2} = A\left(\frac{dy}{dt}\right)+[B \sin(Cy)\times\cos(Dt)]-E \sin(2Cy)$ A,B,C,D are non-zero constants and ...
0
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0answers
11 views

Differentiation of multivariable functions(inverse and chain rule) [duplicate]

In my case i have the following functions: $f(x_1,x_2) = \begin{bmatrix} x_1*x_2^2 + x_1^3*x_2\\x_1^2*x_2 + x_1 + x_2^3\\\end{bmatrix}$ $g(u) = \begin{bmatrix} e^u \\ u^2 + u\\\end{bmatrix}$ An i ...
6
votes
4answers
831 views

Why doesn't derivative difference quotient violate the epsilon-delta definition of a limit?

So the difference quotient is defined as: $$\lim \limits_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ So if we take a function such as $f(x)=x^2$ and go through the simplification, we get $$\lim \limits_{h \...
0
votes
1answer
38 views

Finding Partial Derivative in two ways

I am supposed to find $f_x(0,0)$ of $\frac{5x^2y}{x^4+y^2}$, EDIT: which has a defined value of $0$ at $(0,0)$. The way I did it, I first found the general expression for $f_x(x,y)$, which is $$f_x(...
2
votes
1answer
32 views

Find Polynomial of order 10 for $f(x)=sin(x)$ near x=0

My work so far : I presume the answer should look more like a summation? Thanks!
1
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5answers
98 views

With the linear approx. of $f(x)= sin(x)$ around $0$ Calculate $\lim_{\theta\to 0} \frac{sin\theta}{\theta}$

With the linear approximation of $f(x)= sin(x)$ around $0$, calculate: $$ \lim_{\theta\to 0} \frac{\sin\theta}{\theta}$$ Figured I have to use L'Hospital's Rule, but I think I don't get how to ...
0
votes
0answers
26 views

Which functions can be meaningful differentiable?

I have two functions: $f(x_1,x_2) = \begin{bmatrix} x_1*x_2^2 + x_1^3*x_2\\x_1^2*x_2 + x_1 + x_2^3\\\end{bmatrix}$ $g(u) = \begin{bmatrix} e^u \\ u^2 + u\\\end{bmatrix}$ The questions is:which of ...
1
vote
1answer
23 views

Find intersection angle of curves : $y=x^3-5, y=5x^2-x$

My work so far looks as above. Calculated angle seems too large? Not sure what's wrong in here with my calculations. Thanks!
-1
votes
2answers
42 views

Real analysis: Continuity and Differentiability [closed]

Let $f(x)=x^2$ if $x$ is rational and $f(x)=0$ if $x$ is irrational. a) Prove that f is continuous at exactly one point, namely $x=0$. b) Prove that f is differentiable at exactly one point, namely $...
3
votes
1answer
69 views

Statements about derivatives and integrals [on hold]

My professor gave me one example. It's given one intervall $I=\left [ a,b \right ]\subset \mathbb{R}$ and one function $f:I\mapsto \mathbb{R}$. There is also given 8 statements about derivatives ...
-1
votes
2answers
81 views

If $\lim_{x\to\infty}f(x)=0$, does $\lim_{x\to\infty}f'(x)=0$ as well? [duplicate]

I've been learning about the Fourier transform, and when you apply to derivatives. All of the notes I've read seem to imply that if $$\lim_{x\to\infty}f(x)=0,$$ then $$\lim_{x\to\infty}\frac{df}{dx}(...
0
votes
3answers
29 views

Rate of change of the slope of the tangent

I have this question: Find the rate of change of the slope of the tangent of the function $f(x)=-x^3$ at $x=8$ there are two solutions and i can't decide which is correct. 1- I'll find the second ...
0
votes
1answer
31 views

Find the derivative of $x(t) = \int_0^t \lambda^{t-\tau} y(\tau) d\tau$ in one step

Given $$x(t) = \int_0^t \lambda^{t-\tau} y(\tau) d\tau$$ where $\lambda \in \mathbb{R}_{>0}$ Find $\dot x(t)$ Claim: The answer can be obtained in one step yielding $\dot x = y - \log(1/\...
1
vote
3answers
47 views

How to derive through a convolution?

Let $f(t) = \alpha e^{-\beta t}$, where $\alpha, \beta$ are constants Let $g(t) = y(t)$ Then the resulting convolution $f\ast g$ is: $$f \ast g = \int_0^t \alpha e^{-\beta (t-\tau)} y(\tau) d\tau$$...
4
votes
3answers
76 views

Compute $\frac{d^ny}{dx^n}$ if $y = \frac{7}{1-x}$

I am wondering what is $\frac{d^n}{dx^n}$ if $y = \frac{7}{1-x}$ Basically, I understand that this asks for a formula to calculate any derivative of f(x) (correct me if I'm wrong). Is that related to ...
0
votes
0answers
53 views

Does $f'(x) =0$ imply $f(x) = const$ for every domain?

For $f : [a;b] \rightarrow \mathbb{R}$ $f'(x) =0$ implies $f(x) = const$ but what when domain is not an interval?
2
votes
4answers
53 views

Derivative of exponent

Looking to solve : $$ \frac{d}{dx}[2^{0.5x}]$$ The multiplication and X value in the exponent is confusing me. Help? Thanks!
5
votes
1answer
78 views

What does it take for a smooth homeomorphism to be a diffeomorphism?

I have an open subset $A$ of $\mathbb{R}^k$ and a subset $B$ of $\mathbb{R}^n$, $n>k$, that are homeomorphic and $f:A\longrightarrow B$ is a smooth homeomorphism between two sets. I'm wondering if ...
-1
votes
2answers
50 views

Find the antiderivative…

Find the complete solution of the given differential equation $${dy \over dx} = {3x \sqrt{1+y^2} \over y}$$ I know how to solve it if the right side didn't contain either $x$ or $y$, but I can't ...
0
votes
1answer
47 views

Find $\frac{dy}{dx}$ when $t=0$ for $\begin{cases}x = t^2 + 2t \\ y = 2t^3 - 6t\end{cases}$

A dot is moving on a grid following this rule: $$\begin{cases}x = t^2 + 2t \\ y = 2t^3 - 6t\end{cases}$$ I need to find $\frac{dy}{dx}$ when $t =0$. It seems like I should use implicit ...
0
votes
1answer
24 views

Partial derivatives and differentiability, continuity

Function $f : \mathbb{R}^3 \rightarrow \mathbb{R}$ has in every $x$ of domain partial derivatives $\frac{\partial f}{\partial x_1}(x) =x_2$, $\frac{\partial f}{\partial x_2}(x) =x_1$, $\frac{\partial ...
0
votes
3answers
42 views

Is $|xy|$ differentiable in $(0,0)$?

Is $f(x,y) = |xy|$ differentiable in $(0,0)$? I have no idea how to approach this problem.
0
votes
0answers
15 views

Transformation of an equation in total derivatives

When I studied calculus many years ago, transforming $f'=df/dt$ into $df=f'\cdot dt$ was always accompanied by words of caution that this is a bit lazy. We used it mainly during integral calculation ...
2
votes
1answer
35 views

Differentiation under the integral sign, where the partial derivative of the integrand is not bounded by a Lebesgue integrable function.

Let $K(t)=\int_1^\infty u(t,x)\ \mathrm{d}x$, where $$u(t,x)=\frac{\cos{tx}}{x^2}\mathbb{1}_{[1,\infty)}(x).$$ I need to show that, for $t>0$, $$\frac{dK}{dt}(t)=\frac{1}{t}\left(K(t)-\cos{t}\right)...
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0answers
14 views

Resolution function explicity [on hold]

Examine where the equation $f(x,y)=0$ locally by $y=h(x)$ can be resolved. Calculate in all these places $h'(x)$ by implicit differentiation. Enter the resolution function(s) $h(x)$ explicitly if this ...
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0answers
35 views

Please prove the following question from isc class12 math book [duplicate]

Let $$y=\sqrt {1-\sin 2x\over 1+\sin 2x}$$ , prove that $${dy\over dx}+\sec^2 (\pi/4-x)=0$$
-5
votes
1answer
111 views

Derivative of $\sqrt {1-\sin 2x\over 1+\sin 2x}$ [closed]

Let $$y=\sqrt {1-\sin 2x\over 1+\sin 2x}$$ , prove that $${dy\over dx}+\sec^2 (\pi/4-x)=0$$
2
votes
0answers
70 views

Any reason not to define a derivative as the average of the derivatives on all sides?

We all know $\operatorname{abs}$ is not differentiable in a classical sense, but one question that's always bothered me is, why not define the derivative as the average derivative in each direction? i....
0
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0answers
13 views

Starting from an expression of E(V) and P(V), is there a way of obtaining an expression for E(P)?

I have a function, an equation of state named the Birch-Murnaghan's equation of state, in which the Energy ( $E$ ) is a function of the Volume ($V$), where $E_{0}$, $V_{0}$, $B_{0}$ and $B_{0}^{'}$ ...
1
vote
1answer
44 views

Differentiation under integral sign- Multivariable case problem

Let $f_{\theta}(x,y)=f(x\cos \theta-y\sin \theta,x\sin\theta+y\cos\theta)$, where $f\in C^2(\Bbb{R}^2)$(Is the range necessarily $\Bbb{R}^2$? This is quite ambiguous.) a function with a bounded ...
8
votes
3answers
650 views

Why is/isn't the derivative of a differentiable function continuous?

I am confused about the following Theorem: Let $f: I \to \mathbb{R}^n$, $a \in I$. Then the function $f$ is differentiable in $a$ if and only if there exists a function $\varphi: I \to \mathbb{R}^n$ ...
1
vote
1answer
52 views

A differentiation with first principles question for two variables

I know this question is probably quite easy but it's been some time since I've done any sort of calculus and since a google search failed to turn up anything relevant to this specific question I ...
1
vote
0answers
19 views

Definition of the left and right derivative.

The definition of the derivative is $$g'(a)=\lim \limits_{\delta \rightarrow 0} \frac{g(a+\delta) - g(a)}{\delta}$$ also the left derivative is $$ \lim \limits_{\delta \rightarrow 0^-} \frac{g(a+\...