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

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4
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1answer
343 views

Differentiation under the Integral Sign

Let $X$ be an open subset of $\mathbb{R}$, and $Y$ be a measure space. Suppose that a function $f:X\times Y\rightarrow \mathbb{R}$ satisfies the following conditions: 1.$f(x,y)$ is a measurable ...
2
votes
0answers
17 views

Real Analysis Differential functions

I am currently working through an exercise set and I am a little stuck on the following question: For $a > 0$, define a function $f_a(x)= \begin{cases} x^a \sin(1/x), &x \ne 0\\0, &x=0 ...
0
votes
0answers
12 views

Application of derivatives using particle movements

I have the values with me. But I am not quite sure how to use them. I pulled out a graph but it still made no sense to me. In question B, what do they mean exactly? A particle moves along a straight ...
3
votes
4answers
22 views

derivative of $\ln((1+\beta)^x-1)$

How do I differentiate the term $\ln((1+\beta)^x-1)$ with respect to $x$? Is it possible to do it this way: $$\frac{1}{(1+\beta)^x-1}$$ But i get stuck if i do the normal differentiation.
0
votes
1answer
24 views

Differentiating both sides of an equality with respect to first variables? (Not answered)

I am proving a statement and the truth of the following proposition would help me with it. If anyone could say whether the proposition is true and give a hint how to prove it I would be very much ...
1
vote
2answers
23 views

Use the definition of derivative to prove ln(x+1)/x =1

How would I use the definition of derivative to prove lim (as x->0) [ln(1+x)]/x = 1? I got to [ln(1+x+h)/(x+h) - ln(1+x)/x]/h but have no idea where to go from here. On another site I found ...
0
votes
0answers
21 views

Elementary Analysis, 3rd root question

Prove that $\forall a \in \mathbb{R}$ there is a unique solution to $x^3 = a$ Prove that $\forall x,a \in \mathbb{R}$ $$(x^{1/3}-a^{1/3})(x^{1/3})^2 + a^{1/3} x^{1/3} + ((a^{1/3})^2)=x-a$$ Prove ...
17
votes
5answers
252 views

Geometric interpretation of mixed partial derivatives?

I'm looking for a geometric interpretation of this theorem: My book doesn't give any kind of explanation of it. Again, I'm not looking for a proof - I'm looking for a geometric interpretation. ...
0
votes
1answer
18 views

Which of the following options are correct?

$f(x),g(x)$ are defined on $[-1,1]$, $f'(0),g'(0)$ exist, $f(0)=g(0)$, and $f(x)\ge g(x)$ holds for an open interval containing $0$. Then which of the following is correct: I, $f(x)$ and $g(x)$ have ...
1
vote
1answer
18 views

Why does gradient descent make sense?

Suppose I define two functions of $x$ in terms of a convex function $f$ with a unique minimum $x_0$: $$f_1(x) = 1 \times f(x)$$ $$f_2(x) = 2 \times f(x)$$ Suppose I wanted to minimize each of these ...
2
votes
1answer
39 views

If $\lim_{t\to\infty}\varphi(t)=x_0$, does this imply that $\lim_{t\to\infty}\varphi'(t)=0$?

Let $\phi:\mathbb{R} \to \mathbb{R}^n$ and $\lim_{t \to \infty} \phi(t) = X_0$, where $X_0$ is a constant in $\mathbb{R}^n$ then $\lim_{t\to \infty} \phi'(t) = 0$. I search everywhere and I need to ...
0
votes
0answers
13 views

Matrix derivative involving inverse

Suppose we have a symmetric matric $\boldsymbol{A}$ and its inverse is given by $\boldsymbol{A}^{-1}$. Then, how can we compute the derivative $$ ...
0
votes
0answers
11 views

ODE - Laplace transform

I have an ODE $\psi^{'}(s)_{3 \times 3}=(A+Bs)_{3 \times 3}\psi(s)_{3 \times 3} \tag1$ where A,B are constant skew symmetric matrices with zero determinant. $\psi(s)$ is a rotation matrix. It implies ...
-1
votes
2answers
22 views

Derivative limits exercise [on hold]

Let $f: \mathbb{R} \to \mathbb{R}$ which is differentiable for every $x\in \mathbb{R}$. Prove that $$\lim_{h \to 1} \frac{f(hx) - 2f(x) + f(\frac{x}{h})} {h-1}= 0, \quad x \neq 0$$
1
vote
1answer
23 views

Application of the mean value theorem for Integrals

Suppose that $f(x)$ is a differentiable function in $[a,b]$, $f^{'}(x)$ is a monotone decreasing function in $(a,b)$, and $f^{'}(b)>0$. So how to prove that $$ \big \vert \int_a^b \cos ...
2
votes
0answers
158 views
+200

Infinite Series -: $\psi(s)=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+.+.+ $.

We have a given converging series using derivatives and matrices(Analogue to Taylor's series) $\psi(s)_{3 \times 3}=\psi(0)+\psi_1(0)s+\psi_2(0)\frac{s^2}{2!}+\psi_3(0)\frac{s^3}{3!}+..+.. \tag 1$. ...
1
vote
2answers
112 views

Differentiating with respect to the limit of integration

I'm confused about problems involving differentiation with respect to the limit of an integral, I just want to check that my understanding is correct. For example, are the following statements ...
1
vote
2answers
28 views

derivative of $\frac{2}{3}x^{3-e}$

Find the derivative:$\;\;\;\;\;\;\dfrac{2}{3}x^{3-e}$ I am not sure how to solve this problem. My try: $\ln y=\dfrac{2}{3}(3-e)\ln x$ $\dfrac{1}{y}\times y\;'=\dfrac{2}{3}(3-e)\dfrac{1}{x}$ ...
2
votes
1answer
143 views

Find $f$, such that $\,f,f',\dots,f^{(n-1)}\,$ linearly independent and $\,f^{(n)}=f$

I am trying to find a function $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{C})$, satisfying the differential equation $$ f^{(n)}=f, $$ and with $\,f,f',\dots,f^{(n-1)}\,$ being linearly independent. ...
4
votes
2answers
30 views

Making a piecewise function continuous and differentiable at point

Problem: Let $f(x) = \left\{ \begin{array}{lr} \frac{\arctan(x)}{(1+x)^2} & : x \geq 0\\ Ae^x + B & : x < 0 \end{array} \right. $ Find $A$ and $B$ such that the function is ...
1
vote
2answers
13 views

Critical points for undefined fraction on closed interval

I am told to find the absolute extrema of $$h(x) = \frac{8+x}{8-x},[4,6]$$ So I obtain the derivative of $$\frac{16}{(8-x)^2}$$ The trouble I am having is trying to determine the critical points. ...
0
votes
2answers
32 views

derivative of $e^{\ln x^2}-3x^7$

$$e^{\ln x^2}-3x^7$$ The first term: $=e^v$ $v=\ln x^2=u^2$ $v\;'=2uu\;'=(2\ln x)\dfrac{1}{x}=\dfrac{2\ln x}{x}$ $\dfrac{e^{\ln x^2}2\ln x}{x} +21x^{-8}$ How do I simplify further? I don't ...
0
votes
2answers
15 views

derivative of $y=\sqrt{10^{5-x}}=u^{1/2}$

$y=\sqrt{10^{5-x}}=u^{1/2}$ $y\;'=\dfrac{1}{2}u^{-1/2}\times u\;'=\dfrac{1}{2}(10^{5-x})^{-1/2}=\dfrac{1}{2\sqrt{10^{5-x}}}\times 10^{5-x}\ln10(-1)$ ...
1
vote
2answers
33 views

Evaluating $(\frac{\cos x}{1-\sin x})^2$

$(\dfrac{\cos x}{1-\sin x})^2$ $f\;'(x)= 2(\dfrac{\cos x}{1-\sin x}) \times (\dfrac{-\sin x+\sin^2x-\cos^2x}{(1-\sin x)^2})$ Does $\sin^2x-\cos^2=1$? or $-1$? Then it could factor with the ...
0
votes
0answers
30 views

Derivations on the space of triangular matrices

I have started to research matrices and have been asked the following. If $d$ is a a derivation on $T_n(\mathbb R)$ and $d(e_{ij})=0$, with $1\le i \le j \le n$, Show that for every $r \in ...
1
vote
2answers
31 views

How to find the derivative of $f(x)=(x^3-4x+6)\ln(x^4-6x^2+9)$?

Find the derivative of the following: $$f(x)=(x^3-4x+6)\ln(x^4-6x^2+9)$$ Would I use the chain rule and product rule? So far I have: $$\begin{align}g(x)=x^3-4x+6 \\g'(x)=2x^2-4\end{align}$$ would ...
0
votes
2answers
36 views

Derivative of integral?

When asked questions of the type; What is the derivative of $f(x) = \int_0^{x^2} \frac{cos(t)}{t+1}dt $ ... what is the general method to solve them? Above is just an example from my workbook. I ...
0
votes
2answers
35 views

Derivative Help: $f(x) = x^3\,e^{5x-7}$

I need to find the derivative of the following function: $${\rm f}\left(\,x\,\right)= x^{3}{\rm e}^{5x - 7}$$ but I don't know where to start with this problem. Please help.
1
vote
4answers
28 views

Evaluating $\frac{d}{dx}\sqrt[4]{\ln(12-x^2)}$

Find Derivative and evaluate at $x=1$: $$ \frac{d}{dx}\sqrt[4]{\ln(12-x^2)} = (\ln u)^{1/4} $$ $$v=(v)^{1/4} \implies v=\ln\;u, v\;'=\dfrac{1}{u}(u\;')$$ $$y\;'=\frac{1}{4}v^{-3/4}\; \times ...
0
votes
0answers
17 views

A question about a change of variable

I have came across this question while trying to find the derivate of the inverse functioin. And I have found the following limit: $$ \lim_{y\to y_0} = \frac{1}{\frac{f(x) - f(x_0)}{x-x0}}$$ We also ...
13
votes
2answers
368 views
0
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0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
1
vote
3answers
242 views

Derivative of $(\ln x)^{\ln x}$

How can I differentiate the following function? $$f(x)=(\ln x)^{\ln x}.$$ Is it a composition of functions? And if so, which functions? Thank you.
0
votes
1answer
32 views

Find the equation of normal line to the graph $y=2(x-1)^3$

Find the equation of normal line to the graph $y=2(x-1)^3$ at the point where $x=\frac12$. So far, I found the derivative: $$\frac{dy}{dx}= 6(x-1)^2 $$ What to do next?
2
votes
1answer
678 views

Minimizing L1 Regularization

I have given a high dimensional input $x \in \mathbb{R}^m$ where $m$ is a big number. Linear regression can be applied, but in generel it is expected, that a lot of these dimensions are actually ...
2
votes
1answer
10 views

Help with Implicit Differentiation: Finding an equation for a tangent to a given point on a curve

When working through a problem set containing Implicit Differentiation problems, I've found that I keep getting the wrong answer compared to the one listed at the back of my book. The problem is ...
1
vote
1answer
38 views

Smoothing Lemma

Given a $C^0$ function $g:[a,b]\to \mathbb{R}$ that is smooth everywhere except at $c$ (where $a<c<b$), and has positive derivative everywhere except at $c$, the claim is that there exists a ...
1
vote
4answers
57 views

Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$

How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$. Please help me. Thanks!
2
votes
3answers
36 views

evaluating derivative of $\log_4(2x^2+1)$

Find the derivative and evaluate at $f\;'(2):$ $$\log_4(2x^2+1)$$ $\log_4(2x^2+1)=y$ $4^y=2x^2+1$ $4^y\ln4 \times y\;'=4x$ $y\;'=\dfrac{4x}{4^y\ln4}\implies \dfrac{4x}{(2x^2+1)\ln4}$ What ...
1
vote
3answers
28 views

Derivative of $e^\sqrt{4x+4}$

$$f(x)=e^\sqrt{4x+4}$$ $f(x)=e^u$ $u=\sqrt{4x+4}=(4x+4)^{1/2}$ $u\;'=\dfrac{1}{2}(4x+4)^{-1/2}=\dfrac{1}{2\sqrt{4x+4}}$ I don't know how to proceed from here. Thanks.
1
vote
0answers
15 views

Prove that the evaluation map $E_{x_0}: C(K) \to \mathbb{R}$ is differentiable

Let $K \subset \mathbb{R}^m$ be compact, and pick any $x_0 \in K$. Show that the function $E_{x_0}: C(K) \to \mathbb{R}, E(f) = f(x_0)$ is differentiable. For functions between Euclidean spaces, I'm ...
0
votes
1answer
17 views

Derivative involving inner product

How would I take the derivative of a function $$f(x) = < x,x >=x^{T}x?$$ The answer seems to be 2x but I don't know how to explicitly show this other than saying "there are 2 x's being operated ...
0
votes
2answers
22 views

Showing that a multivariable function is one to one

I am stuck with the following problem I am given the function $f$ such that $f(x,y)=(x^2-y^2,2xy)$ I am supposed to show that the function is one to one. For a function to be one to one, $f'>0$. ...
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votes
1answer
22 views

How to find equations of tangent lines to the graph passing through a line

How to find equations of tangent lines to the graph $f(x)=x/(x-1)$ passing through point $(-1,5)$? Progress I used the quotient rule and got $f'(x)=-1/(x-1)^2$, but I have no idea how to continue.
0
votes
1answer
20 views

Is this function of 2 variables differentiable?

$f(x,y) = \frac{\sin(x^4+y^4)}{x^2+y^2}$ when $(x,y) \neq (0,0)$ and $0$ when $(x,y) = (0,0)$ Is f differentiable?
-2
votes
1answer
27 views

How to use quotient rule to differentiate $f(t)=\frac{\cos t}{t^3}$? [on hold]

The function is $\displaystyle f(t)=\frac{\cos(t)}{t^3}$, and I want to know how to differentiate it using the quotient rule. Thank you so much!
1
vote
2answers
21 views

Chain rule for multiple variables?

What I've tried so far: $$F(x,y,z(x,y)) = 0$$ $$\implies \frac{\partial F}{\partial x} = 0$$ By the chain rule: $$\frac{\partial F}{\partial x} = \frac{\partial F}{\partial z}\frac{\partial ...
0
votes
1answer
21 views

Geometric interpretation of derivative?

For some function $F(x,y) = 0$, $$\frac{dy}{dx} = \frac{-F_x}{F_y}$$ Can someone give me a geometric interpretatio of this? ($F_x$ and $F_y$ are the partial derivatives)
1
vote
3answers
135 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 $a$ ...
-3
votes
1answer
31 views

If $y = 2\sin(x)-\sin^2(x)$ and $x = 2\cos(x)-\sin(x)\cos(x)$ what is $\frac {dy}{dx}$? [on hold]

If $y = 2\sin(x)-\sin^2(x)$$\ \ \ x = 2\cos(x)-\sin(x)\cos(x)$ What would $\frac {dy}{dx}$ equal to? so $\frac {dy}{dx}=2\cos(x)-\frac {2\cos(x)\sin(x)}{-2sin(x)}$ ... ? what would $y'$ of ...