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

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Proving the chain rule

First see the first comment on this post how to prove the chain rule? and this post in general Chain rule proof doubt So we begin by proving the chain rule by assuming we have $f,g$ where $f$ is ...
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3answers
63 views

How do you implicitly differentiate $y$ from $y\sqrt{x^2+y^2} = 15$?

I've been working on this problem for the last 45 minutes, and I keep getting the wrong answer, no matter what I do. I tried squaring the whole equation, so that there was no radical to deal with - ...
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19 views

Wronskian for n-dimensional systems of ODEs

We define Wronskian for the case of a single 2nd-order ODE as $W(y_1, y_2)=\begin{vmatrix} y_1(x) & y_2(x) \\ y_1'(x) & y_2'(x) \end{vmatrix}$. But for the more general case, that is for the ...
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111 views

Finding root of $\cos(x)$ by Newton-Raphson method

The exercise asks me that if I want to find the root of $f(x) = \cos(x) = 0$ using Newton-Raphson method, does the initial value matters? I know that Newton-Raphson method is a special case of the ...
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27 views

Find the points on $y = 1/(2x-1)$ where the slope of the tangent line is $-2$

I have a homework question that I don't know how to approach. Can I get some help? Thanks Find the points on $\displaystyle y = \frac{1}{2x-1}$ where the slope of the tangent line is $-2$.
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41 views

$g'(x) = cg(x)$. Are there any other functions, aside from ${e^{cx}}$, that satisfy the condition?

Assume that $g:\mathbb R \longrightarrow \mathbb R$ and $g'(x) = cg(x)$, where $c\in\mathbb R$ and $\forall\ x\in\mathbb R$. Are there any other functions, aside from ${e^{cx}}$, that satisfy the ...
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1answer
107 views

Differentiability of $f(x) = \exp(-1/x^2), f(0) = 0$ [duplicate]

Let $f: \mathbb{R}\to \mathbb{R}$ be defined by $f(x) = \exp(-1/x^2)$ if $x\neq 0$ and $f(0)=0$. I'm trying to show that $f$ is differentiable with continuous derivative in all points $x\in \mathbb{R}$...
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3answers
79 views

how to find the derivate of a function g(x)

It's $g(x)={{x^{2}-1}\over{x^{2}+2}}$ and i have to calculate $g^{13}(0)$. I can't calculate all the derivates so i think to use power series. $g(x)={{x^2\over{x^{2}+2}}-{1\over{x^2+2}}}$ Can i use ...
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56 views

Analytic functions equal to all orders in a point are equal on the open interval

Let $A\subset \mathbb{R}$ be open. To make everything clear, my definition of analytic function here is: A function $\psi : A\to \mathbb{R}$ of class $C^\infty$ is said to be analytic if for each $...
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1answer
33 views

The second derivative of the composition with a product, $H(x)=h(h(x)h(x^2))$

Let $$H(x)=h(h(x)h(x^2))$$ What is a second derivative of this function? I got first derivative. $$(h'(x)h(x^2)+2xh'(x^2)h(x))+h'(x)h(x^2)$$ Is it right?
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1answer
35 views

Derivative of derivative squared with respect to same variable

What is the answer, and how should I go forth do derivate something like this. $$\frac d{dx}((\frac {dy}{dx})^2)$$
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47 views

How to determine whether or not the partial derivatives are continuous at a given point?

I've been given the following function of two variables: $$f(x, y) = \frac{xy \left(2x^2 - y^2 \right)}{x^2 + 2y^2}$$ and was asked to determine whether or not $f_x(x, y)$ and $f_y(x, y)$ are ...
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1answer
36 views

Differential inequality transformed into an integral bound

Assume that all of the following functions and expressions are defined in such a way that the following make sense. Let $ f:[0,1] \to \mathbb R $ and set $ g(t):= - \log f(t)$ and $ h(t):= f(t)^{1/n}$...
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1answer
42 views

Exchange order of partial differentiation and integration.

Consider a two-valued function $f(x,y) : R^2 \rightarrow R$. Define $f_x(x,y) = \frac{\partial f(x,y)}{\partial x}=\lim_{\epsilon\rightarrow 0}\frac{f(x+\epsilon,y)-f(x,y)}{\epsilon}$ and $f_y(x,y)$ ...
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40 views

relationship between functional derivative and simple derivative.

Consider a function $f(x)$. The derivative of $f$ with respect to $x$ is defined as $$ f'(x)=\lim_{\epsilon\rightarrow 0} \frac{f(x+\epsilon)-f(x)}{\epsilon}. $$ Sometimes, we define a functional ...
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1answer
66 views

Prove Nth derivative expression by induction

I am given the function $f(x) = \sqrt{3x+5}$ I have calculated the expression of the nth derivative to be $$f^{(n)}(x)=\frac{(-1)^{n+1}\cdot(2n-3)!!}{2^n}\cdot(3x+5)^{-(2n-1)/2}\cdot3$$ How would I ...
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1answer
96 views

Prove n-th derivative function

Let $f(x)=(3x+5)^{1/2}$. Obtain and prove a formula for the $n$-th derivative $f^{(n)}$ I may need to find some derivatives of the function, and prove by induction, but I do not know how to do these. ...
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22 views

Question about finding a function $f(x)$ that fits the following bounds

I am in search of a function $f(x)$ that is infinetely differentiable (that is $f^{(n)}(x) \ne 0$ and that the function is defined when differentiated $n$ times) and an interval $[a,b]$ that will ...
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35 views

Show that fraction of differential equation solotions is constant

Let $x_1,x_2,x_3$ be solutions of $\dot{x}(t)+a(t)x(t)=b(t)$ show that$$ \frac{x_2-x_1}{x_3-x_1}= const$$ I tried to isolate x and got to $\frac{x_2-x_1}{x_3-x_1} $ $= \frac{\dot{x_1}-\...
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27 views

derivative with using chain rule

Let $H(x) = h(h(x)h(x^2))$ be. Is it something like this $h'(h(x)h(x^2))(h(x)h(x^2))'?$ what is a derivative of this function?
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1answer
64 views

Show $g(x) = \frac{f(x)-f(x_0)}{x-x_0}$ is continuous

Suppose that $f:(a,b) \to \mathbb{R}$ is continuous on $(a,b)$ and differentiable at $x_0 \in (a,b)$. Define $g(x) = \frac{f(x)-f(x_0)}{x-x_0}$ for $x \in (a,b)\setminus \{x_0\}$ and $g(x_0)=f'(x_0)...
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1answer
65 views

Vector Calculus Notation for “Gradient of a Vector”

Given (differentiable) functions $\,n_{1,2}:\mathbb{R}\to\mathbb{R}\,$ we write vector $\renewcommand{\arraystretch}{2}$ \begin{align} \vec{\boldsymbol{n}} = \begin{bmatrix} n_{1} \\ n_{2} \end{...
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4answers
64 views

How to Completely Simplify the Derivative of $\sqrt{16-x^2}-4\cos^{-1}(x/4)$

I am trying to completely simplify the derivative of the following function: So far, I have gotten the answer: Apparently this is not simplified enough. Does anyone know how to simplify this ...
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2answers
42 views

How to Find the Derivatives of $\tan^2(x^4)$ and $\sec^3(x^5)$?

I am to find the derivative of f(x) and g(x): So far, I know the following: The derivative of tan(x) = sec(x)^2 The derivative of sec(x) = sec(x)tan(x) So, I have tried the following steps to ...
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1answer
35 views

Derivatives, slopes of parallel lines and Related Rates (High School)

I'm a bit confused about some derivative and related rates stuff. I guess it's more like I'm not sure if I've done them correctly: 1.) The diameter of a tree was 12 in. During the following year, the ...
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34 views

How is the derivative of $x[4\sin(2x)+6\cos(2x)]$ the expression $(4-12x)\sin(2x)+(8x+6)\cos(2x)$

I am wondering because I have tried to answer this question, but have gotten a different answer: $(4-6x)\sin(2x)+(4x+6)\cos(2x)$. To get the above answer I did the following steps: 1) Product rule: $...
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24 views

Proof regarding homogeneous functions of degree $n$

If $f$ is homogeneous of degree $n$, show that $f_{x}(tx,ty) = t^{n-1}f_{x}(x,y)$. My attempt at a solution: Let $u = tx$ and $v = ty$.$$\frac{\partial f(u,v)}{\partial x} = \frac{\partial f(u,v)}{\...
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47 views

How to Differentiate $x^7(7x+5)^6$

I am trying to differentiate $f(x) = x^7(7x+5)^6$. So far I have done the following steps: 1) Use the product rule, which is $(x^7(6(7x+5)^5))+((7x^6)(7x+5)^6)$ 2) Factor out $x^6$ and $(7x+5)^5$ ...
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1answer
39 views

Proving invariance along solutions

Consider the equation $$\frac{d^{2}}{dt^{2}}\phi+\frac{g}{L}\sin\phi=0$$ $L>0$ and $g>0$ is the gravity constant. Take $x_{1}=\phi$ and $x_{2}=\frac{d}{dt}\phi$, then we have $$-\frac{g}{L}\cos ...
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397 views

We can define the derivative of a function whose domain is a subset of rational numbers?

Usually the derivative is defined for a function $f:A\to \mathbb{R}$ where $A \subset \mathbb{R}$, and the usual definition of the derivative at a point $a$ require the existence of an open ...
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102 views

Differential Notation Magic in Integration by u-Substitution [duplicate]

I'm really confused now. I always thought that the differential notation $\frac{df}{dx}$ was just that, a notation. But somehow when doing integration by u-substitution I'm told that you can turn ...
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1answer
136 views

Clarification if a disconnected function has a derivative at defined points.

I know so far for a derivative to exist. -The point should not exist as a discontinuity -It should not have a vertical tangent -There should be no sharp corner/ cusp at the point $$(-2)^x=\begin{...
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1answer
113 views

If one of the Dini derivatives is bounded, then f is Lipschitz

If one of its Dini derivatives (say $D^+$) is bounded show that a function $f$ satisfies a Lipschitz condition, Definition of the upper right Dini derivative: $$D^{+}f(x) = \limsup_{h\to\ 0^+} \...
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30 views

Find the critical numbers of $f(x) = 7x^3 + |x|$. Determine critical numbers at which the tangent line is horizontal

Find the critical numbers of $f(x) = 7x^3 + |x|$. Determine critical numbers at which the tangent line is horizontal Here is what I have so far: I know that $|x|$ can be rewritten as $\sqrt{x^2}$ ...
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1answer
22 views

Definition of Concavity for Twice Differentiable Functions

Let $f(x)$ be a twice-differentiable function. The definitions for concave upward and concave downward I found in my textbooks are all somewhat wordy, something along the lines of: Definition 1: $...
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2answers
66 views

Who invented the notation $Df$ for the derivative?

We are often taught that $f'$ came from Newton and $\frac{df}{dx}$ came from Leibniz, but who introduced $Df$? Are there other notations for this simple idea by famous mathematicians?
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78 views

Finding first, second and third derivative

How would I find the first, second and third derivatives of the function $$f(x)=f\left(f(x)f\left(x^2\right)\right)\ ?$$
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1answer
136 views

nth derivative of a radical function [duplicate]

Let $f(x) = \sqrt{3x+5}$. Obtain and prove a formula for the nth derivative. I'm having trouble finding the formula for the nth derivative. I've computed the first three derivatives but not really ...
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1answer
65 views

Can we draw the graph of the derivative/integral of a function by using the graph of the function only?

Consider a function say $F(x) = x^2 + 5\sin x$ then we have it's derivative as $F'(x) = 2x + 5\cos x$ and thus we have the graph of $F'(x)$ quiet easily but can we plot a graph using only the graph of ...
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121 views

How to compute the $n_{th}$ derivative of a composition: ${\left( {f \circ g} \right)^{(n)}}=?$

I know that there is a general formula to compute the $n_{th}$ derivative of a product which is as follows $${\left( {fg} \right)^{(n)}} = \sum\limits_{k = 0}^n {\left( {\begin{array}{*{20}{c}} n\\ k ...
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1answer
36 views

A converse of Lagrange Mean value theorem?

Let $f:[a,b] \to \mathbb R$ be continuous function , diferentiable in $(a,b)$ and $c \in (a,b)$ be such that $f'(c)$ is not a local extreme value of $f'(x)$ in $(a,b)$ and $c$ is not an accumulation ...
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1answer
59 views

Landon derivation of the Gaussian distribution

This is the follow up to the question Taylor series of a convolution. Continuing the derivation given at Probability Theory: The Logic Of Science By E. T. Jaynes, chapter 7 "The Central Gaussian, Or ...
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1answer
59 views

$f:[a,b] \to \mathbb R$ be continuous , differentiable function such that $f'$ is bounded on $[a,b]$ ; then is $f' \in \mathcal R[a,b]$?

Let $f:[a,b] \to \mathbb R$ be continuous , differentiable function ( having one sided derivatives at $a$ and $b$ ) such that $f'$ is bounded on $[a,b]$ ; then is $f'$ Riemann integrable on $[a,b]$ ?
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125 views

Derivative of a negative order?

Below, $\Delta$ means taking the derivative, $\frac{d}{dx}$. For $n\in\mathbb{Z}$, $n\geq 0$, we have $$\Delta^n\sin{x}=\sin{(x+n\tau/4)} \\ \Delta^n\cos{x}=\cos{(x+n\tau/4)}$$ I found that out while ...
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31 views

Is IERS/Rapid Service/Prediction Centre wrong?

At http://www.iers.org/SharedDocs/Publikationen/EN/IERS/Publications/ar/ar2012/ar2012_352.pdf?__blob=publicationFile&v=1 page 80 it is stated: "The LOD for the combination are derived ...
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1answer
34 views

Definition of determinant of a derivative.

Can someone please help me with the following definition: $B$ is a bounded open set in $R^n$ and $g$ : $\bar{B} \rightarrow R^n$ is $C^1$ . We say $a$ is a regular value of $g$ if $\det{(g'(x))}\...
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1answer
23 views

differentiability of bizarre piecewise function

Let $T = \{3k \mid k \in \Bbb Z \}$ and $f : \Bbb R \to \Bbb R$ be defined by $f(x) = \left\{ \begin{array} {ll} x^2, && x \in T \\ x, && x \notin T \end{array} \right.$. Show that $...
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176 views

how to find increasing/ decreasing intervals based on graphs

Use the given graph of the derivative f ' of a continuous function f over the interval (0, 9) to find the following. graph On what interval(s) is f increasing? (Enter your answer using interval ...
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132 views

Finding intervals of a contentious function

The graph of the derivative f ' of a continuous function f is shown below. (Assume f ' continues to ∞.) GRAPH (a) On what interval is f increasing? (Enter your answer in interval notation.) On what ...
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30 views

Derivative of a definite integral from a parameter

Is there any way to calculate $a$ to satisfy $$\frac{\partial{F(a)}}{\partial{a}}=0,$$ where $$F(a)=\int_{-\infty}^{+\infty} f(x,a)dx$$ $f$ can be any function but we know that the above ...