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

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2answers
117 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 ...
4
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2answers
32 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
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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. ...
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2answers
33 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 ...
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2answers
20 views

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

$y=\sqrt{10^{5-x}}=u^{1/2}$ $y\;'=\dfrac{1}{2}u^{-1/2}\times u\;'=\dfrac{1}{2}(10^{5-x})^{-1/2}=\dfrac{1}{2\sqrt{10^{5-x}}}\times 10^{5-x}\ln10(-1)$ ...
2
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2answers
34 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 ...
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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 ...
1
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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
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2answers
36 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.
0
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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 ...
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 ...
0
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1answer
33 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
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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 ...
2
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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 ...
1
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3answers
29 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.
2
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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
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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 ...
1
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4answers
29 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
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1answer
18 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
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2answers
28 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, $f'>0$. ...
-1
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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
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1answer
21 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
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1answer
28 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
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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 ...
1
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4answers
58 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!
0
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1answer
22 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)
-3
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1answer
31 views

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

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 ...
0
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3answers
19 views

How to find the points where the slope of the tangent is $-1$?

For the function $f(x) = x^3 - 4x$, find the points where the slope of the tangent is $-1$. Use the algebraic method. Do I need just to find zeros?
2
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1answer
31 views

Differentiability of the convolution $\int_0^tf(t-s)g(s)\;ds$

Given two continuously differentiable functions $f,g:[0,\infty)\to\mathbb{R}$. I want to know what we can tell about the differentiability of $$(f\ast g)(t)=\int_0^tf(t-s)g(s)\;ds$$ Especially, why ...
0
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1answer
32 views

Second derivative of $\sec(3x)\sqrt{324\cos^2(3x) + 396 + 121\sec^2(3x)}$

How to take second derivative of $$\sec(3x)\sqrt{324\cos^2(3x) + 396 + 121\sec^2(3x)}.$$ I am having trouble with taking the second derivative of this. I know I should simplify it before taking the ...
0
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3answers
40 views

Evaluating $\frac{\operatorname d \! \phantom x}{\operatorname d\!x}\frac{4}{\ln(x^2+2)}$

$\dfrac{\operatorname d \! \phantom x}{\operatorname d\!x}\dfrac{4}{\ln(x^2+2)}= \dfrac{4}{\ln u}$ $u=x^2+2$ $u\;'=2x$ $y\;'=\dfrac{4}{\dfrac{1}{u}} \times (u\;') \implies ...
0
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4answers
42 views

Using parametric differentiation for $\frac{\operatorname d \! y}{\operatorname d \!x}$?

Hi so I'm in my calculus class and the teacher gave us a problem to do. I'm not quite sure how to attack this question. He's given us a couple of steps but I don't understand. If someone can further ...
0
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3answers
21 views

Derivative of $\frac{d}{dt}\ln(6t^2+9t+12)=$

$\dfrac{d}{dt}\ln(6t^2+9t+12)=$ $y=2\ln(6t)+\ln(9t)+\ln(12)$ $y\;'=2\dfrac{1}{6t}(6)+\dfrac{1}{9t}(9)+0$ $=\dfrac{12}{6t}+\dfrac{9}{9t}=\dfrac{2}{t}+\dfrac{1}{t}$ What am I doing wrong?
-1
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1answer
18 views

Finding the max with velocity and acceleration graph

I'm confused on why there is a maximum at R. If I flipped the acceleration graph it looks like a continuously increasing function with no max or min to me. Could someone help me understand this? 8a ...
0
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1answer
20 views

Some question about definition of derivative

We can alternatively write the definition of derivative at $x_0$ by $\lim_{h \to 0}\frac{f(x_0+h)-f(x_0)}{h}$. Can we say that if $h=x/M$, $\lim_{M \to \infty}\frac{f(x_0+x/M)-f(x_0)}{x/M}$? Assume ...
1
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1answer
39 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 ...
0
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1answer
19 views

Implicit Differentiation of $\cos^3(y)$

I think I understand implicit differentiation outside of those problems involving trig functions but for some reason this problem is breaking my brain: Assume that y is a function of x. Find ...
2
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3answers
39 views

derivative problem. is it same?

First derivative of $y=\ln(x)^{\cos x}$ is $-\sin x\ln x+\frac{\cos x}{x}$ or another answer? My friend gets another answer, but it's true? thanks.
1
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0answers
72 views

how to solve this limit with $e^{x}$

I was trying to solve the derivative of $e^{x}$ the traditional way with the definition of the derivative: $$ \lim_{h\rightarrow 0}\frac{e^{x+h}-e^{x}}{h} $$ so I solved like this: ...
1
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1answer
24 views

Deriving of the Jacobi bracket and the chain rule

This is from a passage that derives the Jacobi bracket from first principles. I cannot understand how the first equality works. It seems to use the chain rule and I agree with the second term but ...
0
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1answer
28 views

Calc I limit/series question

Let $f : \mathbb R\rightarrow\mathbb R$ be a function that is differentiable at zero and such that $f(0)=0$. Show that for each $n\in \mathbb N$ we have that ...
0
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2answers
33 views

Parametric form of square

What is the appropriate parametric equation of the boundary of a square? For example, the unit circle has a parametric equation $x(t)=\cos(t)$ and $y(t)=\sin(t)$.
3
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1answer
35 views

Failing to reproduce specific Functional derivative

I'm failing to reproduce an (indirect) result in a paper, namely $${δF[g]\overδg(x,y,z)}={r^4\over\ell^5} $$ where $F[g]=\iiint \frac{2dxdydz}{\ell g(x,y,z)} $ and $g(x,y,z)=-{\ell^2 \over r^2} $. ...
1
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1answer
21 views

Schwarzian derivative of inverse function.

Let $\mathcal{D}$ denote the Schwarzian derivative. I have to prove that if $\mathcal{D}f(x)$ exists $\forall x$ then $\mathcal{D}f^{-1}$ exists $\forall x\in D_{f^{-1}}$ then find a formula. I tried ...
0
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2answers
35 views

Derivative of a arctan

I am not sure on how to even start, I know how to do simpler derivative but not a complex one like this. I need to find $\frac{d(x^2 arctan(5x))}{dx}$
0
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2answers
23 views

Trouble finding second implicit derivative

I have trouble finding the second implicit derivative. This is the question. Find y'' in terms of x and y by implicit differentiation. $x^5 +y^5 = 2^5$ The final answer I always get is ...
1
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1answer
25 views

Definition of a functions with respect to partials

I am stuck with the following problem: I am given that $$F(x,y)=f(x,y,g(x,y)) =0.$$ I am asked to show $D_1g$ and $D_2g$ with respect to the partials of $f$ My idea was to write that $DF=DfDg$ ...
1
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1answer
16 views

Calculus Question ( Surface area/Volume)

An open box (I.e. no lid) had a square base of side $x$ cm and height $h$ cm. Given that the volume of box is $108$ cm$^3$. a) Show that the surface area in cm$^2$ is given by $A = x^2 + 432/x$. b) ...
2
votes
1answer
17 views

Incongruencies with derivatives and differencials

I read in Piskunov that the increment $\Delta y$ of a function can be written as: $\Delta y = f'(x) \Delta x + \alpha \Delta x$ And, when ${\Delta x\to 0}$ , $dy=f'(x)dx$ The problem is, doesn't ...
2
votes
2answers
24 views

Can someone explain how to calculate the third order partial derivative of $f.$

$f(x,y)=\sin(xy).$ I calculated that $ \dfrac{\partial^2f}{\partial x\,\partial y}=\dfrac{\partial^2f}{\partial y\,\partial x}=\cos(xy)-xy\sin(xy)$. I also calculated $$ \frac{\partial^3f}{\partial ...