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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1answer
20 views

Find the derivative using the definition derivatives: $f(x) = xH(x)$ where $H(x) = 0$ when $x < 0$ & $ H(x) = x$ when $x\ge 0$

My solution: the derivative doesn't come out as nicely as I had thought, can someone verify that I'm solving this properly? Using the definition: lim $H(x)_{x\to a} = (f(x)- f(a))/(x-a)$ Which ...
4
votes
2answers
36 views

Show Laplace operator is rotationally invariant

I'm trying to show the Laplace operator is rotationally invariant. Essentially this boils down to showing $$\frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} = \frac{\partial^2 ...
1
vote
1answer
22 views

Differential equation of inclined plane

I'm having some trouble with the equation $$\frac{d}{dt}\dot{x}=g\sin\Theta \implies \dot{x}(t)=\dot{x}(t=0)+\int_0^t dt'\:g\sin\Theta=\dot{x_0}+g\:t\sin\Theta $$ which appears in page 4 of ...
0
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0answers
32 views

Parametric Equation of a parabola from the derivative of the parametric equation of a circle

Find the velocity and trajectory to throw a ball from a Ferris Wheel to a friend standing below. The Ferris Wheel has a diameter of 16 meters and its highest point is 19 meters above the ground. It ...
5
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2answers
48 views

Mean Value Theorem: finding two numbers in the same interval

Let $f,g:[0,1] \to \mathbb{R}$ such that $f'(x)>0,\ g'(x) >0,\ \forall x\in [0,1]$. Moreover, $f(0)= g(0)$ and $f(1)=g(1)$. Prove that exists $x_1, x_2 \in [0,1]$ such that $$f(x_1) = g(x_2), \ ...
0
votes
2answers
26 views

find the gradient of the curve at the points where it crosses straight line

Find the gradient of the curve $y = x \sqrt{4-x^2}$ at the points where it crosses the straight line $y=x$ I've calculated $f'(x) = \frac{4}{\sqrt{4-x^2}}$, but I'm not clear what to do next.
3
votes
1answer
83 views

$f $ once differentiable does $ f_{xy}$ exist?

Suppose $f:{\bf R}^2 \rightarrow {\bf R}$ is once differentiable at a point $p$. Does it follow that $f_{xy}$(the derivative of $f $ w.r.t to $x$ and then w.r.t to $y$) exist at $p$?
0
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1answer
17 views

Different quotient of $f(x,y,z,h) = \int_{0}^{\infty}e^{-u/2}(h+u)\frac{3x(h+u+z)}{(x^{2}+y^{2}+(h+u+z)^{2})^{\frac{5}{2}}}du$ with respect to $h$

I have to compute the limit of the difference quotient of the function $f(x,y,z,h)$ defined as: $f(x,y,z,h) = \int_{0}^{\infty}e^{-u/2}(h+u)\frac{3x(h+u+z)}{(x^{2}+y^{2}+(h+u+z)^{2})^{\frac{5}{2}}}du$ ...
0
votes
4answers
52 views

Derivative of $y = \cos^2(x^3 + x^2)$

So the problem I am stuck on is this: find the derivative of $$y = \cos^2(x^3 + x^2)$$ I am very lost in all of this, so please explain the steps, that would be a great help.
1
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2answers
32 views

Differentiability - general function?

I know that a function is differentiable if the limit exists as $\Delta x \to 0$ of a certain limit. But how can one know this beforehand? I mean, we usually just differentiate using rules that we ...
1
vote
4answers
38 views

Criterion to satisfy Rolle's Theorem.

$f(x) = \begin{cases} x^a\log x, & \text{if $x \neq 0$,} \\[2ex] 0, & \text{if $x=0$. } \end{cases} $ What should be the value of $a$ so that f satisfies Rolle's theorem in [0,1] ?? What I ...
0
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1answer
24 views

one to one of multivariate function…

Suppose $f: \mathbb{R}^k \to \mathbb{R}^k $ has positive definite gradient matrix, $\dot{f}\equiv ( \partial f_i/\partial x_j )$. Then, where can I refer to see that $f$ is 1-1 or how to prove it? ...
0
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0answers
10 views

Visualisation of gradient and computation?

I am learning differentiability in several variables, and I am stuck. I cannot visualize the definition that $\lim: \lim_{f(a+h)-f(a)-ch/h\to 0} = 0$ The book indicates that $ch$ is called gradient. ...
1
vote
1answer
37 views

Derivative of the Square Root of a Logarithm

$f(x)=ln(\sqrt {\frac {6x+3}{3x-9}})$ Find $f'(x)$. I've tried the method involving distributing the natural log and making it $(1/2)\ln(6x+3)-(1/2)\ln(3x-9)$.
2
votes
2answers
29 views

Find the value of $x$ for which the gradient is zero.

Given that $y=(x-3) \sqrt{x-1} $. Find the value of $x$ for which the gradient is zero. The derivative of the function is $$\frac{dy}{dx} =\frac{x-3}{2\sqrt{x-1}} + \sqrt{x-1}.$$
-1
votes
3answers
49 views

The curve given by $y = \sqrt{x} \,(x-4)^4$ [closed]

A curve is given by $y = \sqrt{x} \,(x-4)^4$. I need to find $\dfrac{dy}{dx}$, and then identify the $x$ co-ordinates of the points where $\dfrac{dy}{dx} = 0$. I'm not sure how to differentiate the ...
0
votes
1answer
22 views

Solving coupled non-linear equations

I am struggling to understand what the following question requires me to do: I believe I need to differentiate implicitly, but am unsure how I show it cannot be done.
1
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2answers
31 views

Property of a function $f$ satisfying $|f(x) - f(y)| \leq (x - y)^2$

I am given a relation about differentiable function that $$|f(x)-f(y)| \le (x-y)^2$$ and if $f(0)=0$ then possible value of $f(1)=?$ I saw the slope form and rewrote it as $$ \frac{|f(x)-f(y)|}{x-y} ...
1
vote
2answers
53 views

Use the formal definition of the derivative to find the derivative of $f(x) = \frac1x$

Working through some exercises which I have been set in a Stats module. I'm stuck on this problem. I can get to $$\lim_{h\to0}\frac{\frac1{x+h}-\frac1x}h.$$ Then I'm unsure as to where to go from ...
0
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0answers
29 views

n integrals in summation

I've seen it might be possible to write a summation that looks like $$ \sum\limits_{i=1}^{\infty}\left\{\frac{\partial}{\partial x_i}\left(\frac{xy}{\sqrt{x^2+y^2}}\right)\right\} $$ But what about ...
1
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1answer
16 views

to find in which directions does Derivatives at a point exists?

If suppose I have a fuction e.g. $f(x,y,z)=|x+y+z|$ ,and I'm asked to prove that in which directions does derivative of $f$ at a point,(say $e_1-e_2$) it exists. How to think about the problem ?
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3answers
136 views

determine whether $f(x, y) = \frac{xy^3}{x^2 + y^4}$ is differentiable at $(0, 0)$.

I am new to multivariable calculus and my textbook doesn't give out solutions so I'm just wondering how you go about proving something like this? I know that a function is differential at a point $a$ ...
0
votes
1answer
33 views

Solve the following derivative through its definition

We have a function $$f(x) = \frac{5}{\sqrt{x} + 1}$$ and its definition states that $$f'(x) = \lim_{x \to 0}\frac{f(x+h)-f(x)}{h}.$$ Therefore, I attempted it by computing the following $$\lim_{x ...
0
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1answer
40 views

Help with derivative inside a summation

I have $\sum_{k=0}^{\infty}k^2q^kp=\sum_{k=0}^{\infty}k[kq^{k-1}]qp=\sum_{k=0}^{\infty}k[\frac{d}{dq}(q^k)]qp$. How can I go about pulling this $\frac{d}{dq}$ outside of the sum?
1
vote
2answers
39 views

Product rule proof for $f,g: U\subset\mathbb{R}^n \to \mathbb{R}$

I've been struggling with that proposition but I don't know how to prove it. Let $f: U\subset\mathbb{R}^n \to \mathbb{R}$, $g: U\subset\mathbb{R}^n \to \mathbb{R}$ be functions that are ...
2
votes
2answers
50 views

Minimum volume cone.

What would be the radius and the altitude of a right circular cone that circumscribes a sphere with a radius 8 cm if the volume of the cone is to be minimized? Here is my rough sketch; My idea is ...
2
votes
2answers
184 views

Shorthand notation for partial?

If I am taking a regular derivative, and I want to show the process in detail, I'll do something of the sort $f'(x) = g'(x) + h'(x) - l'(x) ..... $, etc, using that "prime" notation. However, what ...
0
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0answers
13 views

Integration : Green's Function in estimating displacement of non-prismatic beams

I'm working on a non-prismatic structure similar to that in Figure 3 of Page 10 (345) from an article entitled: "Green’s function for the deflection of non-prismatic simply supported beams by an ...
0
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1answer
58 views

complex differentiation

It is my first text here. So I have started to look at complex numbers in death. I do Uni know, so adding $3+4i$ and $4+7i$ is now nothing. What I am stuck on is the idea of taking a derivative of a ...
1
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2answers
59 views

How to find sum of coefficients of $\frac{d^n}{dx^n} \left( Z(x)^m \right)$

I noticed something interesting result but I do not know how to prove it (or disprove) Function $U$ defined as $$ U(Z(x),Z'(x),Z''(x),Z'''(x),...,Z^{(n)}(x))=\frac{d^n}{dx^n} \left( Z^m(x) ...
2
votes
3answers
43 views

differentiation of the following equation 3

i already done the differentiation, just wanna confirm either i got it right or wrong. Can someone verify this for me. 1) f(x) = $ -3\over x^{5/2}$ f '(x) = $ 3({ 5\over 2}x^{3/2})$ . ...
1
vote
1answer
34 views

how to solve $f''(e^{x} \cdot \sin x)$

How do I find the derivative $f''(e^{x} \cdot \sin x)$. I start to find $f'(x)$ by using the product rule $f'(e^{x} \cdot \sin x) = e^{x} \cdot \sin x + e^{x} \cdot \cos x = e^{x}(\sin x + \cos x)$ ...
1
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2answers
33 views

Epsilon-delta proof with $x$ and $y$ defined

I am stuck with the following problem. The question is as follows: prove that for all $x$ in $[0,2]$, there exists $y$ in $[0,2]$ such that the function $f(x,y)=0$. The function $f$ is defined as ...
0
votes
1answer
21 views

Implicit Differentiation, Plugging in Values

Suppose that $(g(x))^2+16x=x^2g(x)+1$ and that $g(4)=7$ Find $g'(4)$ So for the derivative, I got $(2xg(x)-16)/(2g(x)-x^2)$ After plugging in the values, I got $\frac{10}3$ for my answer. Could ...
1
vote
1answer
42 views

John Lee's Intro to Smooth Manifolds Inverse Function Theorem

In John Lee's "Intro to Smooth Manifolds" Chapter 7, p 160, we have a proof of the inverse function theorem. Here, in the middle of the page we have $F_2 = DF(0)^{-1} \circ F$. Is $DF(0)^{-1}$ a ...
0
votes
2answers
32 views

Use $\lim_{x\to0} \frac{\sin x}{x} = 1$ to evaluate these limits

Use $\lim_{x\to0} \frac{\sin x}{x} = 1$ to evaluate the limits: a) $$\lim_{x\to0} \frac{x\tan^2(x)}{\cos(3x)\sin^3(2x)}$$ b) $$\lim_{x\to \frac{\pi}{2}} \frac{\tan(2x)}{x-\frac{\pi}{2}}$$ Can ...
2
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2answers
43 views

Why is the second derivative test inconclusive for some local max/mins?

I know what the second derivative test is, when it is can be used, and when it can't. So I am not asking any of those questions. What I am asking is why we could have a local max at $c$, have ...
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0answers
27 views

Differential equations, derivative of determinant, Euler's formula

Let $b:\mathbb{R}^n\to\mathbb{R}^n$ be a smooth vector field. Let $u(s,x,t):\mathbb{R}^{n+2}\to\mathbb{R}^n$ with $s,t\in\mathbb{R}$ and $x\in\mathbb{R}^n$ satisfy the following differential ...
0
votes
2answers
28 views

Find the derivative using its definition

Use the definition of the derivative to find $f'(x)$ if $f(x) = \frac{3}{x^{0.5}+2} , x>0$ To begin with the definition is $$f'(x) = \lim_{h \rightarrow 0}\frac{f(x+h)-f(x)}{h}$$ Thus, this is ...
0
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2answers
22 views

Find the equation of the tangent line in which the point is not on the graph

Given the function $f(x) = \dfrac{(x-1)}x$, find the equation of the tangent line to the graph of $f$ that pass though the point $(4,1)$. NOTE: The point (4,1) is NOT on the graph of f. Okay so first ...
0
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0answers
26 views

Calculating derivatives in high school calculus

I have this problem in my homework and I am a little confused on the presentation of the derivative. Can this be interpreted as $$P(V) = \frac{7}{V} = 7V^{-1}?$$ So $$P'(V) = -7V^{-2}$$ (Using the ...
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0answers
16 views

A plane flying with a constant speed of 14 km/min passes over a ground radar station at an altitude of 15 km and climbs at an angle of 40 degrees.

A plane flying with a constant speed of 14 km/min passes over a ground radar station at an altitude of 15 km and climbs at an angle of 40 degrees. At what rate, in km/min, is the distance from the ...
0
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2answers
74 views

Calculus related rate question!

I'm really stump at this question, asking for instantaneous rate of change! A visitor of the Jurassic Park attraction lost his way and is now walking to the East from point $A$. At the same time, a ...
2
votes
2answers
36 views

How to derive $J_v(x)$

I've seen many sources say that $\frac{\text{d}}{\text{d}x}J_v(x) = J_{v-1}(x) - \frac{v}{x}J_v(x)$, but every time I try to derive it, I get the conjugate, $\frac{v}{x}J_v(x) - J_{v-1}(x)$. Could ...
0
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0answers
10 views

Derivatives of the Fourier transform vanishing on a countable set: construction

Can we construct a time-limited function $f(t)$ whose Fourier transform $F(\omega)$ has the following property: for a given $\omega_0 \in \mathbb{R}\backslash\{0\}$ and $N\in\mathbb{N}$, we have (1) ...
0
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1answer
43 views

Why does the order not matter? Partial D

When taking partial derivatives, why does the order not matter as long as the function is continuous? Any proof, intuitive or rigorous?
0
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0answers
24 views

Checking this function for differentiability

$f(x) = |x|\sin x + |{x-\pi}|\cos x$ for $x \in \mathbb{R}$ Is the above function differentiable at $x=0$ ? At $x=\pi$ ?
2
votes
3answers
60 views

Can't get this implicit differentiation

I've been working at this implicit differentiation problem for a little over an hour now, and I, nor my friends can figure it out. The question reads "Find the equation of the tangent line to the ...
3
votes
2answers
36 views

Why is the derivative of the arccos the negative derivative of arcsin?

$$ \dfrac{d}{dx} \sin^{-1}x = \dfrac{1}{\sqrt{1-x^2}}$$ $$\dfrac{d}{dx} \cos^{-1}x = - \dfrac{d}{dx} \sin^{-1}x$$ What is the reason for this?
2
votes
0answers
42 views

Why do we need partial derivatives?

Partial derivatives are used to find the slope of a three dimensional curve at an angle. But why do we need them? Can't another function be created from the first one which calculates $x$ and $y$ ...