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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3answers
20 views

Why does $\frac{1}{6e^{2y}}=\frac{1}{2x-8}$ in this context?

This is the context: I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
0
votes
0answers
17 views

Points of contacts of tangents of the curve $y=\sin x$

Prove that the points of contacts of tangents of the curve $y=\sin x$ drawn from origin lie on the curve $1/x^2 -1/y^2 = -1$
7
votes
1answer
89 views

Function that is the sum of all of its derivatives

I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful ...
1
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1answer
10 views

$\frac{d}{dy} F(g(y),y) = ?$

Given that we know if the integral of $f(y)$ is $F(y)$ then we can say that $\frac{d}{dy} F(y) = F'(y) = f(y)$. But what does $\frac{d}{dy} F(g(y),y)$ equal to? Can we say that it is $f(g(y),y)?$
-1
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2answers
20 views

Need help in verifying if I am taking the derivative of $f(x) = \frac{x}{\cos(x)}$ correctly

I need to take the derivative of $f(x) = \frac{x}{\cos(x)}$. What I am doing: $$f'(x) = \frac{d\ (x\cos(x)^{-1})}{d \ x} + (\frac{d\ (x\cos(x)^{-1})}{d\ (\cos(x))} * \frac{d\ \cos(x)}{d\ x})$$ ...
0
votes
0answers
10 views

Problem in proving that covariant derivative of a vector transforms as a tensor.

I have got this extra term while trying to prove the tensor nature of the covariant derivative of a vector. $\frac{\partial y^r}{\partial x^k} \frac{\partial^2 x^j}{\partial y^r \partial y^m} v'^m(y) ...
0
votes
1answer
33 views

If $f$ is differentible at a point $x \in [a,b]$, then $f$ is continuous at $x$.

Proof. As $t\rightarrow x$, we have, by Theorem 4.4, (Baby Rudin, p.104) $f\left( t\right) -f\left( x\right)$ = $\dfrac {f\left( t\right) -f\left( x\right) } {t-x}\cdot \left( t-x\right) \rightarrow ...
0
votes
1answer
22 views

Differentiating a function composition

Given $g:R^n \rightarrow R^k$ and $h:R^k \rightarrow R$, we have $f(x) = h(g(x))$. Using the chain rule, we can differentiate $f(x)$ to get $f'(x) = \nabla^Th(g(x))g'(x)$ My question is why do we ...
0
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0answers
13 views

Relation between sub-tangent and sub-normal of $y^2 = (x+a)^3$

If the relation between sub-tangent and sub-tangent at any point on the curve $y^2 = (x+a)^3$ is $p(SN)=q(ST)$ then find the value of $p$ and $q$ where $SN$ is length of sub-normal and $ST$ is length ...
1
vote
4answers
69 views

Proving that the exponential inequality $e^x \ge x^e$ holds for all $x \ge 0$ [duplicate]

How does one prove that $$e^x \ge x^e$$ for all $x \ge 0$? I tried to do this by setting $f(x)=e^x-x^e$ Plotting this function shows this easily, as seen here. However, when I tried to prove ...
0
votes
0answers
41 views

Bound the first derivative of the following function: $f(x)=g(x)+h(x)$

Consider a decreasing function $f(x)$, resulting from the sum of two other decreasing functions: $f(x)=g(x)+h(x)$. All these 3 functions are positive. In addition, we have $g(0)=h(0)=1$. Further, we ...
1
vote
2answers
43 views

What does third derivative tell about inflection point?

I was trying to find the nature (maxima,minima,inflection points) of the function $$\frac{x^5}{20}-\frac{x^4}{12}+5=0$$ But I faced a conceptual problem.It is given in the solution to the problem ...
7
votes
1answer
63 views

Find all $f:\mathbb {R} \rightarrow \mathbb {R}$ where $f(f(x))=f'(x)f(x)+c$

Recently, while studying calculus, I have come across multiples problems which asked the following: If $f(x)$ is a polynomial, find all $f(x)$ that $f(f(x))=f'(x)f(x)+c$, where $c$ is a constant. ...
-1
votes
0answers
22 views

which of the following functions are differentiable at x=0 [on hold]

The function is $$f(x) = \arctan \left( \frac{1}{|x|} \right), \quad( x\ne 0), \quad f(0) = \frac{\pi}{2}$$ i have found its left derivative -1 and right derivative 1? but it is derivable. how?
0
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1answer
30 views

Why does this follow from the triangle inequality?

Proving that differentiability implies continuity.
0
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3answers
29 views

What does this notation mean on this derivative question?

$x^2 \tan x \mid _{x=\pi}$ What does this little line after the $\tan x$ mean? Does it mean I should take the derivative and then substitue in $\pi$? I need to find the second derivative.
1
vote
1answer
47 views

How do you differentiate the integral from $ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt$ [duplicate]

How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t? $$ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt $$ I know the answer is $$ e^x\sqrt{1+e^{2x}} + ...
1
vote
1answer
16 views

Error Bounds with Trapezoidal Formula

I know there are some posts about the same thing but I am unable to do my specific question or at least, I don't think I'm doing it the right way. The question asks me to use the Trapezoidal Error ...
5
votes
4answers
88 views

application of L'Hopital's rule?

I am trying to evaluate the following limit: $$ \lim_{x \to 0} \frac{e^x}{\sum_{n = 1}^\infty n^k e^{-nx}}, $$ where $k$ is a large (but fixed) positive integer. I am unsure how to proceed. Can this ...
3
votes
1answer
56 views

A general case - Leibniz rule

Given the Leibniz rule: $$\frac{d}{dy}\int^a_b f(x,y) dx = \int_b^a \frac{\partial f}{\partial y} (x,y) dx$$ How do I prove a more general case using the chain rule and the above: $$\frac{d}{dy} ...
0
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2answers
27 views

How to calculate this derivative? Do I have to use the chain rule?

Derivative of $(r^2 - x^2)^{1/2}$ with respect to x? Please show the steps also.
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2answers
41 views

How to differentiate $y=e^ {\frac{-1}{2}x^2}\cdot\sqrt{1+2x^2}$?

We need to find the maximum point of the curve, M. I know that we must find the $dy/dx$ of the equation of the curve and set it to $0$. However, I'm having trouble differentiating the equation. I'm ...
0
votes
2answers
25 views

On differentiable functions

Consider the integral: $I$= $\int_2^{\infty} f(x)g(x) \mathrm{d}x $, where $f(x)$ and$g(x)$ are nonconstant functions. Assuming that this integral exists, does this necessarily mean that $f(x)$ and ...
0
votes
0answers
6 views

Giving two examples of functions with some properties.

This is a question from a list. Obtain two $\mathcal{C}^\infty$ functions $f,g:\mathbb{R}\to\mathbb{R}$ satisfying these properties: $f(x)=0 \Leftrightarrow 0\leq x\leq 1$; $g(x)=x$ if $|x|\leq 1$, ...
0
votes
2answers
39 views

Complex differentiability and differentiability in R2

In $\mathbb R$ for a derivative to exist (or a limit generally) it is necessary that the limit be the same in both directions (from below and above) and this is the same in $\mathbb C$ where for a ...
0
votes
2answers
27 views

differentiability of a function

Let $f$ be a continuous function on an open interval in $\mathbb R$ such that $|f|$ is differentiable. Can we show that $f$ is differentiable? I can get several examples of non-differentiable $f$ if ...
1
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1answer
13 views

Prove that the equation of the tangent at P is $ \frac {xx_1}{a^2} - \frac {yy_1}{b^2} = 1 $ (Hyperbolas)

Question: Point P ($x_1 , y_1$) is on the hyperbola $\frac {x^2}{a^2}$ - $\frac {y^2}{b^2}$ = 1 Prove that the equation of the tangent at P is $$ \frac {xx_1}{a^2} - \frac ...
0
votes
0answers
31 views

The derivative of a function of multiple variables

I am trying to understand a step in the theory section of my differential equations textbook. The author writes, For example, suppose we transform the first order differential equation ...
5
votes
4answers
85 views

Showing that if $\lim_{x\to\infty}f'(x)=L$ then $\lim_{x\to\infty}\frac{f(x)}{x} = L$.

Let $f:[0,\infty)\to\mathbb{R}$ derivable and suppose that $$\lim_{x\to\infty}f'(x)=L.$$ How can I prove that $$\lim_{x\to\infty}\frac{f(x)}{x} = L$$? I have solved some similar problems using the ...
1
vote
0answers
19 views

Upper bound on hessian

Given a smooth Riemannian manifold $(\mathcal{M},g)$ and $f \in C^{\infty}(\mathcal{M})$ let $r(x)= d(x,x_0)$ where $d$ is the distance function wrt $g$ and $x_0$ is some point on the manifold. If we ...
2
votes
2answers
31 views

Prove a function approaches infnity when the deriviative is greater than $0$

Here's my question: Let $f$ be a function which has a derivative in $\Bbb R$ such that $f'(x)\geq0$ and $f''(x)\geq0$ for all $x \in \Bbb R$ Prove that if there is some $a \in \Bbb R$ such ...
-2
votes
0answers
29 views

Use differentials to approximate $f(x,y)=x^4+2x^2y^2-xy^4 $at $x=10.36$ and $y=1.04$ [on hold]

Hope someone can help with this one. Use differentials to approximate $f(x,y)=x^4+2x^2y^2-xy^4$ at $x=10.36$ and $y=1.04$.
0
votes
0answers
13 views

Limits: Show derivative approaches 0 and infinity.

Consider: $F(K,L) = A[\alpha K^\psi+(1-\alpha)L^\psi]^{1/\psi} $ with $0 \lt \psi \lt 1$ The questions asks to find whether the following two conditions hold: $Lim_{K \to 0}F_k(K,L) = \infty $ and ...
1
vote
1answer
29 views

Derivative of matrix logarithm with respect to matrix

I saw in this post that $\frac{d}{dt}\text{logm}(Z(t)) = \frac{dZ(t)}{dt}(Z(t))^{-1}$ Is this true to say: $\frac{d}{{dU}}{\mathop{\rm logm}\nolimits} (A) = {A^{ - 1}}\frac{d}{{dU}}A$ where U is ...
0
votes
1answer
34 views

Differentiating $- \sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha}\int_0^E\frac{1}{4\pi t}\exp({\omega^2 t - \frac{|x - n - y|^2}{4t^2}})dt$ wrt $x$?

I have a formula for the Ewald method which can be used to speed up computations when working with periodic Green's functions. I will need to take the derivative of the function $G(x, y)$ with respect ...
2
votes
0answers
35 views

How to prove this criteria of differentiability? [duplicate]

Let $f: I \to \mathbb{R}$ continuous and $a\in \operatorname{int}(I)$. Suppose that there is $L\in\mathbb{R}$ such that $$\lim \frac{f(y_n)-f(x_n)}{ y_n-x_n}=L$$ for all sequences $(x_n)$ and $(y_n)$ ...
1
vote
1answer
23 views

Deriving the limit of a sequence?

Consider a function $\mu:\mathbb{R}\rightarrow\mathbb{R}$ differentiable at $0$. Now, consider the sequence $$ \sqrt{n}(\mu(\frac{h}{\sqrt{n}})-\mu(0)) $$ for $n \in \mathbb{N}$, $h \in \mathbb{R}$. ...
0
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0answers
33 views
+50

Show a function is Lipschitz

Suppose a real-valued function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by $$f(x) = \begin{cases} \hfill e^{-\frac{1}{\delta^2 - x^2} + \frac{1}{\delta^2}} \hfill & \text{ $|x| ...
4
votes
1answer
31 views

Does differentiability imply having bounded variation on some subinterval?

Suppose that $f:(a,b)\to\mathbb{R}$ is a differentiable function. Does it follow that $f$ has bounded variation on some subinterval $[c,d]\subset (a,b)$? Details and ideas Being differentiable ...
0
votes
1answer
48 views

$f$ convex strictly decreasing function , is $f'(x+\delta)-f'(x)$ convex

Assume you have a strictly decreasing convex differentiable function $f(x)$, $x \in \Bbb R^+$, I am wondering if the increment of the first derivative is also convex; i.e., $$g(x) = f'(x+\delta) - ...
1
vote
2answers
28 views

Evaluate $\lim_{x \rightarrow 1^+} (\ln\ x)^{x-1}$

Evaluate: $$\lim_{x \rightarrow 1^+} (\ln\ x)^{x-1}$$
0
votes
1answer
30 views

If $(u,v)$ is a point on $4x^2+a^2y^2=4a^2$,where $4<a^2<8$,that is farthest from $(0,-2)$ then $u+v$ is equal to?

If $(u,v)$ is a point on $4x^2+a^2y^2=4a^2$,where $4<a^2<8$,that is farthest from $(0,-2)$ then $u+v$ is equal to? My Approach: I took a parametric point $(t,4-4t^2/a^2)$.And then tried to ...
0
votes
1answer
18 views

How to use Rolle's theorem to verify the following?

How to use Rolle's theorem to verify the location of roots ? $f(x)=x^3+4/x^2+7$ has exactly one zero in ($-\infty$,$0$) I can do it without Rolle's theorem by finding the stationary point which is ...
0
votes
0answers
17 views

Application of average and instantaneous rate of change

John's business is currently selling 175 cookie boxes per day, but sales are dropping at a rate of 2 per day. he is currently charging $6 per box but to compensate for the dwindling sales , he is ...
1
vote
0answers
18 views

Prove $V(x)$ is an increasing function (involving PDF and CDF)

I need to prove the following: $V(x) = x + G(x)/g(x)$ is an increasing function where $G(x)$ is a CDF and $g(x)$ is the corresponding pdf. When I take the derivative, I get $$1 + g(x)^2/g(x)^2 - ...
1
vote
1answer
29 views

Clarke's generalized gradient formula computed on functions defined on open sets

In the book [1], Clarke et al. define the generalized gradient for a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ as follows. 8.1. Theorem (Generalized Gradient Formula). Let ...
0
votes
1answer
11 views

Relation implication

For $n \geq 0$ the following holds: $y^{(n+2)}(0) = -n^2y^{(n)}(0)$ Given the above relation, where superscript denotes the $n$th derivative with respect to $x$ and $(0)$ means function is evaluated ...
0
votes
4answers
74 views

hint to find the second derivative

Let $f:\mathbb{R}\to\mathbb{R}$ be a non-constant, three times differentiable function. If $f\left(1+\frac{1}{n}\right)=1$ for all integers n, then $f''(1)=$? by the given condition for n=0 $f(1)=1\\ ...
3
votes
1answer
31 views

Does continuous extension of a function and its densely defined derivative imply everywhere differentiability?

Let $V \subset \mathbb R^n$ be a closed set, and let $U \subset V$ be open as a subset of $\mathbb R^n$ and dense in $V$. Let $f:V \to \mathbb R$ and $G: V \to \mathbb R^n$ be continuous, with $G = ...
0
votes
1answer
14 views

Correct simultanous application of chain and product rule

For two continuous differentiable functions $g(x)$ and $h(x)$ we seek $$\frac{d}{dx} [g(x) h^{-1}(x)]$$ where $h^{-1}(x) = \frac{1}{h(x)}$. This asks us to apply product and chain rule in sequence, ...