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

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1answer
27 views

Smoothness of component functions

Let $X$ be a real Banach space and $V$ be a real, finite dimensional vector space. Consinder the map $F\colon X\rightarrow V$, $F(x)=\sum_{i=1}^{n}\lambda_i(x)e_i(x)$, where $\lambda_i\colon ...
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3answers
100 views

Anyone who can help me with this one

I have an home assignment where I am supposed to differentiate $$\sqrt{\frac{\cos(f(x))}{\sin(g(x))}}$$ The other expression (I had more than this one) gave me no trouble, but this one is hard. I ...
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4answers
57 views

Tangent to the curve $y=x^2-6x+14$ [closed]

Find value of constant $k$ for which the line $y+2x=k$ is a tangent to the curve $y=x^2-6x+14$. I am very confused. Please help thanks.
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1answer
24 views

Smoothnes of a map between Banach spaces when the pointwise evaluation is smooth.

Let $X$, $Y$ be real Banach spaces. Consider a map $F\colon X\rightarrow C^1(Y,\mathbb{R})$ such that for every $y\in Y$ the map $F(\cdot,y)\colon X\rightarrow \mathbb{R}$ is smooth. Claim: Then ...
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1answer
24 views

Help needed on homogeneous functions

If $f$ is homogeneous of degree $1$, is differentiable at $0$ and $f(0)=0$, how can i prove that: $f(x) = \bigtriangledown f(0)\cdot x$
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2answers
48 views

Derivative of $\int_0^{x^2} \sec(t) \,dt$

How do I derive this? I always get confused since there is $x$ and $t$ involved
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6answers
288 views

How does $\tan^{-1}(x-\sqrt{1+x^2})=\frac{1}{2}\tan^{-1}x+C$ directly?

I'm teaching baby calculus recitation this semester, and I meet a problem to calculate the derivative of $$y=\tan^{-1}(x-\sqrt{1+x^2})$$ Just apply the chain rule and after some preliminary algebra, ...
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1answer
29 views

derivative using chain rule

$h(x)c(y)exp(w(y)t(x))$ the derivative of this with respect to y is $h(x)c'(y)exp(w(y)t(x))+h(x)c(y)w'(y)t(x)exp(w(y)t(x))$ I am having trouble with the derivative of this term because using the ...
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4answers
64 views

Given a derivative function, a coordinate on original function, can we find a certain limit?

True or false? If we're given: $f'(x) = \frac 1x$ and $f(2) = 9$, then (true or false): $$\lim \limits_{x \to 2}\frac{{\sqrt {f(x)}}-3}{x-2}= \frac 1{12}$$ (Apologize for fractions, couldn't get ...
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1answer
34 views

Deriving an equation from known information.

Start with $$m_1 \, \frac{d^2r_1}{dt^2} = F_{12} \qquad m_2 \, \frac{d^2r_2}{dt^2} = F_{21}$$ and derive the equation $$\frac{m_1m_2}{m_1 + m_2} \, \frac{d^2r}{dt^2} = F_{21} $$ where $r_1$ and ...
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0answers
36 views

Finding equation of tangent line when $f '(x) = 1$

Given the equation: $$f(x) = x^4 + 2x^2$$ Find the equation of the tangent line when $f '(x) = 1$. I know to set $f '(x)$ to $1$, but it is the work after that that is giving me trouble.
2
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2answers
61 views

Find the points on the graph of $f(x) = 12(x + 9) − (x + 9)^3$ where the tangent line is horizontal.

I cannot figure out how to get started on this question. Would I First simplify, and then take the derivative? Please help!
0
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0answers
17 views

Second derivative of indetermined function

Suppose we have $g(x)$ and $g'(1)=g''(1)=1$ Now lets define $f(x,y)=xg(\frac{x}{y})$ And it is necessary to find $\frac{\delta^2 f}{\delta x^2}(1,1)+\frac{\delta^2 f}{\delta y^2}(1,1)$ So first I ...
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1answer
37 views

Applying Derivative Rules To Complicated Functions

I am having difficulties with a problem that asks us to take the derivative of a function that has many different functions inside of it. The question is as follows: Let $$f(x) = ...
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0answers
6 views

Order of continuity of the cosine interpolant

I can't found any reference about the order of continuity of the well-known cosine interpolant: (1 - cos(t * PI) / 2; , where t is in the interval [0, 1] Anyone ...
0
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1answer
47 views

Is there exists a $y$ in the interval $(0,1)$ such that $f(y)=f(y+1) ?$

A function $f(x)$ is continuous in the interval $[0,2].$ It is known that $f(0)=f(2)=−1$ and $f(1)=1.$ Which one of the following statements must be true$?$ Options are $:$ There exists a $y$ in ...
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3answers
112 views

Prove the derivative of the natural logarithm using the limit definition.

I know how to prove that the derivative of $\ln(x)$ is ${1\over x}$ using the definition $f'(x) = {f(x+h) - f(x) \over h}$ but I have ran into trouble proving that the derivative of $\ln(f(x))$ is ...
0
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1answer
24 views

Is there a way to simplify this further before deriving?

I'm supposed to find the second order derivative at $x=\pi/16$ for $0<x<\pi/8$. I was wondering if there was a way to simplify this equation further than just factoring out 196 from the square ...
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0answers
12 views

Depth change in a cone

A tank is in the shape of an inverted cone. It is being 􏰃filled with water at a rate of 100 cm2/sec. When the depth is 50 cm and the radius is 25 cm, what is the rate of change of the depth with ...
2
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1answer
45 views

How to find the inflection point of $f(x)=\frac{e^x}{1+e^{2x}}$

With the second derivative, the solution gives logarithm of negative number, but I know the graph and the function changes the concavity. $$f(x)=\frac{e^x}{1+e^{2x}}$$ What are the inflection ...
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0answers
57 views

Differentiation w/ a Cone and Cylinder

I am trying to solve the following problem: https://docs.google.com/document/d/1pibMkkuY_1Onve8WzUBRhxF7_qZlgLyiFAOpjTEnjcI/edit?usp=sharing This is my attempt at solving: Given: Cone: r=2cm.; ...
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2answers
46 views

How to find the particular solution of the differential equation.

Original equation $$x^8y'+7y=e^\frac{1}{x^7}\quad y(1)=e$$ This is the original equation with the initial condition. How would i separate to take the integral? This is what I have done so far but I am ...
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2answers
59 views

Prove derivative is continuous

Suppose $f$ is differentiable on an open interval $I$, $c\in I$, and $\lim_{x\to c} f'(x) = L \neq\infty$. Prove that $f'$ is continuous at $c$. In other words, $f'$ cannot have a removable ...
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1answer
24 views

Differentiating functions

I need to differentiate $f(t) = t\sqrt[4]{t}-\frac{1}{\sqrt[4]{t}}$ $y = \frac{x^{2}-5 x+4}{x^{2}}$ $y= x^{4/3}-8x^{2/3}$ Using the derivative rules, I got $f'(t)=\frac{-1}{4}t^{\frac{-3}{4}}$ along ...
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2answers
25 views

Given a string $ab$ and string length $n$, how can I prove that $|\{ (x,y) : x\text{ is index of } a, y\text{ is index of } b , x < y \}| \le n^2/4 $

This is a coding question that I ran into, basically you have a string: "ababa", you need to find: $$S = \{(x,y): x\text{ is index of }a,\ y\text{ is index of }b, x < y\}$$ $$S = \{(0,1) (0,3) ...
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2answers
12 views

How do we find the value of $x$ where the tangent line to $f(x)$ is horizontal?

Consider the function $f(x)=9x^2 +7x$ The derivative is $f'(x)=18x+7$ The slope of the tangent to the graph of $f(x)$ at $x=2$ is $43$ The equation for the tangent line at $x=2$ is $y=43x-36$ I found ...
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0answers
96 views

Differentiation under the integral sign for an electrostatic field

Let $\rho\in C(\bar{D})$ be a continuous function on the compact set $\bar{D}$ and let us define ...
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0answers
29 views

Why do I fail to get the derivative of sec(x) using the power rule?

So, the power rule seems to state: $\frac{d}{dx}x^n = nx^{n-1}$ Since $\sec{x} = \frac{1}{\cos x}$, and the power rule is true, shouldn't the derivative be: $\frac{d}{dx}\sec{x} = ...
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1answer
38 views

Could we define derivatives from the Taylor expansion?

Would it be equivalent to the usual definition if we defined the derivative $Df$ of a function $f: \text{I} \subseteq R \to \Bbb R$, where $\text{I}$ is some open interval in $\Bbb R$, as the function ...
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2answers
34 views

Extreme Values of $x^2-y^2$ constraint to $x^2+y^2=1$

$f(x,y)=x^2-y^2$ constraint to $x^2+y^2=1$ $f_x=2x$ and $f_y=-2y$ $\implies$ the critical point is at $(0,0)$ However, $(0,0)$ does not occur in the constraint. does that mean i don't have to ...
0
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1answer
194 views

How to find relationship of depth (height) and volume when adding water to a sphere container?

How would I find the formula that relates the height of water to the volume of water in a sphere when the water is added at 0.5cm^3/s and the sphere's radius is 1cm (or if easier, any radius, if not, ...
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1answer
23 views

Vertical Acceleration of a Plane

I was given this homework question over the weekend and I've been racking my brain trying to figure out how to proceed. It's basically calculating different factors of the descent of a plane. This ...
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0answers
29 views

How do I solve this differential equation with the initial conditions?

Solve the differential equation $f′′(x)= sin(x)+cos(x)$ with the initial conditions $f′(π)=2$ and $f(0)=π$. This is on my homework but I've never seen anything like it and it's due on Monday, and we ...
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0answers
37 views

Can this problem be solved without the assumption that f is C^2?

EDIT: I noticed that this question was downvoted twice today, so I decided to edit and add more context, e.g., the full problem statement that I am working on and have proved, although I used an ...
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2answers
30 views

Is it always possible to isolate $\frac {dy}{dx}$ in an implicit equation: $f(x,y)=k$?

Is it always possible to isolate $\frac {dy}{dx}$ in an implicit equation: $f(x,y)=k$? For example: $x^y+\ln(y)=3 \rightarrow y'+\frac y{x \ln x}+\frac {y'}y=0 \rightarrow y'=-\frac y{x \ln x}\cdot ...
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2answers
40 views

Showing that the $n$th derivative of this piecewise function vanishes at $0$

Let $$f(x)= \begin{cases} e^{-1/x} & x>0 \\ 0 & x\le 0 \end{cases} $$ I want to show that $f^{(n)}(0)=0$ for every $n\in\mathbb{N}$. My idea is to do this by induction on ...
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1answer
54 views

What do I do next when trying to find the derivative of this fraction?

I'm trying to find the derivative of this equation: $-\frac{3(x-6)}{2\sqrt{9-x}}$ The quotient rule: $\frac{d}{dx}[\frac{f(x)}{g(x)}]=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}$ where $g(x)$ and $f(x)$ are ...
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2answers
53 views

A Question about derivative.

Suppose that $f:\mathbb{R} \to \mathbb{R}$ is differentiable, $f(0)= 0$, and $f’(x) > f(x)$ for all $x \in \mathbb{R}$. Prove that $f(x) > 0$ for $x >0$. Clear, $f'(0)> 0$ and by ...
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3answers
25 views

Calculating the limit of a trigonometric function

Calculate $$\lim_{x \to \pi /2 }{\frac{\cos x}{x-\frac{\pi}{2}}}$$ by relating it to a value of $(\cos x)' $ My thoughts are to manipulate the limit algebraically and then just solve it. But how ...
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2answers
52 views

Fundamental Theorem of Calculus With Weird Limits

$$f(x) =\int_6^{x^3} \sin^3(5t) \, dt$$ what is $f'(x)$? I know I have to use fundamental theorem of calculus, but what do I do about the $x^3$ and $t^3$?
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1answer
116 views

Finding derivative of $x^3-17x$ using limit definition

Could someone help me with the algebra that comes with finding the derivative of $x^3-17x$ using the limit definition of:
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1answer
19 views

Directional Derivatives With Respect to Negative Vectors

I understand this is probably a silly question but I'm with it struggling nonetheless. Consider the directional derivative of $f(x)=x^2$ at $x=1$ with respect to $u=-1$. I can see that this is equal ...
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1answer
50 views

To prove that $f(x)\ge {{1}\over {2}}(x-1)$

Question : Let $f\gt 0$ be continuous on $[1,\infty )$ . Let $$g(x)= \int_1^x f(t)dt\le [f(x)]^2 . \tag1$$ Prove that $$f(x)\ge {{1}\over {2}} (x-1)$$ My Thoughts : Seeing the square on the ...
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3answers
49 views

Find the derivative of $\int_a^{g(x)} f(t)dt $ wrt $x$

Question : Let $$f:[a,b]\rightarrow \mathbb R$$ be continuous and $$g:[c,d]\rightarrow \mathbb R$$ be differentiable . Define $$\psi(x) := \int_a^{g(x)} f(t)dt $$ . Prove that $\psi$ is ...
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4answers
273 views

A question about existance of derivative of function at Zero

Assume that $f:\mathbb{R}\mapsto\mathbb{R}$ is continuous and differentiable everywhere but at $0$. If $\displaystyle\lim_{x\to0} f'(x) = L$ exists, then does it follow that $f'(0)$ exists? Prove or ...
5
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3answers
198 views

Proving differentiability

I just had a question on proving differentiability by showing that the difference quotient exists. I understand in the case of a function like $f(x)=x^2$, where you end up with $((x+h)^2 - x^2)/h = 2x ...
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1answer
468 views

Is there a function that doesn't have a derivative?

I was wondering if such a function exist. I'm comfortable with derivatives of polynomial functions, and some other basic functions, but I'm wondering if there could exist a very complicated function ...
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2answers
68 views

Functions which are continuous but non differentiable at a point except modulus function

I have noticed one thing during solving problems : That is, wherever I find a function which is continuous but non differentiable at a point, there has always been some |.|(modulus) function or ...
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1answer
24 views

Differentiating norm containing vectors and a matrix

I would like to differentiate $$D = ||L^{-1} (x-y)||_2^{2}$$, while x and y are vectors and L is a matrix Can someone show me how to do this? In other words, how to calculate: $$\frac{dD}{dx}$$ ...
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0answers
25 views

Intuition for grad(F(x)) = F(x) grad(log(F(x))?

It is straightforward to prove that $\triangledown f(x) = f(x) \; \triangledown\log(f(x))$. But is there some intuitive way to understand this identity?