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

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The definition of partial derivative

$$\lim_{\Delta z \to 0}\frac{\left(rN_{Az} \right) |_{z + \Delta z}-\left(rN_{Az} \right)|_z}{\Delta z}+\lim_{\Delta z \to 0}\frac{\left(rN_{Ar} \right) |_{r + \Delta r}-\left(rN_{Ar} ...
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1answer
29 views

Is this function symmetrical?

I have created a function $B_{n,k}(f'(x),f''(x),\cdots,f^{(n-k+1)}(x))_{(f \rightarrow g)^c}$ that behaves as follows: $$ B_{n,k}(f'(x),f''(x),\cdots,f^{(n-k+1)}(x))_{(f \rightarrow g)^c} = ...
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2answers
32 views

Find trig derivative of $y=4x(7x+\cot{7x})^6$

Find trig derivative of $y=4x(7x+\cot{7x})^6$. I got $y'= 4(7x+\cot{7x})^6 + 168x(7x+\cot{7x})^5 (\cot^2 {7x})$ but I'm not sure I did it right. Your help is appreciated (:
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1answer
11 views

Find the point $T(a,b)$ on the curve $y = x^2$ which has the shortest distance between itself and the point $P(3,0)$ [Solution Verification]]

Find the point $T(a,b)$ on the curve $y = x^2$ which has the shortest distance between itself and the point $P(3,0)$. $$ \\ \begin{align} \\ y &= f(x) = x^2 \\ b &= f(a) = a^2 \\ \\ &T(a, ...
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3answers
17 views

Derivate a logaritmic function

Let's take $ f = \ln(x) $. The derivate is $ f' = 1/x$. However $g = \ln(50x) $ has the same derivate $f' = g'$. How come? If I where going to derivate $g$ I would substitute $x$ for $t$: $g = ...
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3answers
21 views

Find the point $T(a, b)$ on the curve $y = x^2 + 1$ whose tangent passes through the point $P(1, 0)$ from the left.

Find the point $T(a, b)$ on the curve $y = x^2 + 1$ whose tangent passes through the point $P(1, 0)$ from the left. $$ \\ \begin{align} \\ \\ f(x) &= x^2 + 1 \\ f(a) &= a^2 + 1 = b \\ \\ T(a, ...
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5answers
154 views

Find the derivative of the function. y = $\sqrt{7x+\sqrt{7x+\sqrt{7x}}} $

This question is really tricky. I am wondering if I am right?
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1answer
60 views

Find the derivative of the function. $y = \cot^2(\sin θ)$

My work is as follows. Criticism welcomed. $$y = \cot^2(\sin\theta) = (\cot(\sin\theta))^2$$ Power Rule combined with the Chain Rule: $$\begin{align} y' & = 2(\cot(\sin \theta)) \cdot \frac ...
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2answers
49 views

Does the function $f(x)=x$, $x\in (0,1)$ have a maximum and minimum value?

My book says that since we cannot determine the value of x when it is just less than 1 and just greater than 0, hence the function does not have a maxima or minima. But the fact confuses me because ...
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1answer
31 views

Related Rates Question With Cylinder?

On a test we needed to solve the following question: A right circular cylinder with a constant volume is decreasing in height at a rate of 0.2 in/sec. At the moment that the height is 4 inches ...
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1answer
86 views

Formula for the $n^{th}$ derivative of $f(x)$

I am presented the following prompt: Find a formula for the $n^{th}$ derivative of $f(x) = \frac{x^n}{1-x}$ I've split the function into two parts to differentiate at the suggestion of some users (I ...
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1answer
32 views

logarithmic differentiation issue

Trying to understand a solution I was given to a problem I was told to use logarithmic differentiation on. $$ 1/x(x+1)(x+2) $$ and I know that $$log((ab)/c) = log(a) + log(b) - log(c)$$ So I tried to ...
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0answers
13 views

In the space of polynomials of degree 2 or less, given the derivative linear transformation D and $T:=1+D+D^2$, $S:=1-D$, show that $S=T^{-1}$

Let $ P_2[X] $ be the space of polynomials of degree equal or less than 2 over the field R. Let: $$ D: P_2[X] \rightarrow P_2[X] $$ Be the derivative linear transformation, defined as follows: $$ ...
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0answers
37 views

proof of a series

Hi cant figure out how to prove the following: given $$ f(x)=(x-b)^5+(x-a)^4 $$ prove that the point $f'(c)=0$ divides the segment $[a,b]$ to a ratio of $4:5$ hint: use Rolle's theorm
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2answers
21 views

Example is required

I am trying to find a seuqence of a continuous functions $\{f_n\}$ defined on $[0,1]$ bounded by some small number, say $\varepsilon$ with the additional requirement of $f_n^\prime(t_0)=1$ at a ...
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1answer
18 views

Find the equation of the normal line to the function.

Here is the problem as well as my work: Am I correct? I am unsure if I correctly related the slope of the tangent line to that of the normal line..
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3answers
235 views

Derivative getting different result

Studying for a midterm, and one of the problems is: $$\frac{x^3+7}{x}$$ and we have to find the derivative. My professor is getting: $$2x-\frac{7}{x^2}$$ But I got $$3x-\frac{x^3+7}{x^2}$$ I even ...
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6answers
274 views

Find the derivative using the chain rule and the quotient rule

$$f(x) = \left(\frac{x}{x+1}\right)^4$$ Find $f'(x)$. Here is my work: $$f'(x) = \frac{4x^3\left(x+1\right)^4-4\left(x+1\right)^3x^4}{\left(x+1\right)^8}$$ $$f'(x) = ...
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1answer
50 views

surfaces, curves and lines

Could someone please assist with the following questions: Consider $f(x,y) = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and take $C$ to be the curve of intersection of $z = f(x,y)$ with the plane $y=x$. Show ...
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3answers
89 views

Differentiate $\,y = 9x^2 \sin x \tan x:$ Did I Solve This Correctly?

I'm posting my initial work up to this point. Criticism welcomed! Using the formula $(fgh)' = f'gh+fg'h + fgh'$, differentiate$$y = 9x^2\sin x \tan x$$ $$\begin{align} y' &= 9\frac ...
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1answer
41 views

When finding the derivative of a function why do we have cancel out the x's in the numerator and the denominator?

I get why we cancel them out but I do not understand why we have to. Take $x^3+\frac{2}{x^2}$ for example. Why is $3x^2+\frac{2}{2x}$ wrong? Note: Please use terminology that someone just learning ...
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1answer
29 views

partial derivative in an exat equation

I have to determinate if this equation is an exact differential equation, but I don't now how get the partial derivative respect X & Y, I am confuse, please help me step by step I would ...
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1answer
33 views

How to calculate a Differentiable Quotient?

This is more than likely an Algebra problem but I can not figure out where the $-4x^2$ came from - first equation -3rd line. I do see that they transferred the $2\sqrt{x}$ to the denominator. What ...
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3answers
37 views

What is meant by “find the slope of the tangent to the graph of f at a general point x”

I am pretty thick and need questions to be specific or I do not know what they want. Do they want me to give a random example for x? eg the slope at x=7 is 5x?
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1answer
29 views

Derivative tests question

Show that $k(x) = \sin^{-1}(x)$ has $0$ inflections $2$ critical points $0$ max/min I find that the first derivative is $$\frac{1}{\sqrt{1-x^2}}$$ Second derivative is ...
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1answer
43 views

Definition of a 2-variable function derivative

I read this definition in a book of multivariable calculus: $f(x,y)$ is differentiable at $(x_0,y_0)$ if it can be expressed as the form $$f(x_0+\Delta x, y_0+\Delta y)=f(x_0,y_0)+A\Delta ...
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1answer
37 views

Differentiability of the function $x \mapsto |x|^{3/2}$ at $x = 0$

Could someone please explain whether the function $$\vert x \vert^{3/2}$$ is differentiable at zero? ($x$ here is a real number.) I tried investigating the right and left-sided limits (i.e., the ...
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1answer
17 views

The existence of the $n$th derivative at $c$ presumes the existence of the $(n-1)$st derivative in an interval containing $c$

The following is from Introduction to Real Analysis by Bartle. If the derivative $f'(x)$ of a function $f$ exists at every point $x$ in an interval $I$ containing a point $c$, then we can consider ...
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3answers
59 views

Calculating arc length $y=x^2$

I picked this example for practice and got stuck with it. Someone moderate me if I am in the right path. I need to calculate the length of arc s, on the section of the curve $y=x^2$ with $0≤x≤1$ My ...
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1answer
23 views

Differentiating hyperbolic functions.

$\DeclareMathOperator{\sech}{sech}$Can anyhow me how to differentiate the following? I already tried using the product rule, but I can't quiet seem to succeed. $\sech^{2} x$. $2\bigl(\cosh(2x) - ...
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1answer
20 views

Differentiability implies continuity in $R^2$

Let F be a function from $R^2$ to $R^2$. F is differentiable at a point (a,b) in $R^2$, prove that F is continuous at this point. Can i write F(x,y)= F(a,b)+ c(x-a)+ d(x-b)+e where c,d,e are real ...
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2answers
42 views

Finding slope at a point in a direction on a 3d surface

This is not a duplicate, I have attempted the question and am not sure why my answer is incorrect. QUESTION: The surface with equation $z = x^3 +xy^2 $ intersects the plane with equation $ 2x−2y = 1$ ...
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1answer
25 views

Showing differentiability for a multivariable piecewise function

Let $$f(x,y)=\begin{cases} xy\sin(x/y) & y\neq 0 \\ 0 & y=0\end{cases},$$ show whether $f(x,y)$ is differentiable at $(0,0)$. It seems that there are multiple ways to do this but ...
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2answers
28 views

Find the absolute min and max in the given intervals

$k(x) = e^{-\frac{x^2}{2}}$ on $[-1,2]$ I think the derivative of that is $ -x e^{-\frac{x^2}{2}}$. I don't know how to find zero from that equation.
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1answer
55 views

Differentiation of the function $\operatorname{li}(x) = \int_2^x \frac{dt}{\ln(t)} $

I have to differentiate with respect to x: $$\operatorname{li}(x) = \int_2^x \frac{dt}{\ln(t)} $$ I havn't come across this before, so my idea is to integrate it first? (Backward right?). let ...
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2answers
28 views

Computing the limit of an integral (Derivatives of Integrals)

Assuming that $f(x)$ is continuous in the neighborhood of $a$, compute $$ \lim_{x \to a} \frac{x}{x-a} \int_a^x f(t)dt $$
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1answer
29 views

How can I approximate a function that is not derivable with derivable ones?

Suppose that I have a function whose graph has many angles (i.e. my function is not derivable). How can I approximate this function with derivable ones? Thank you!
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1answer
18 views

Is there a locally-distance-preserving map projection?

I'm trying to figure out if there is a family of map projections which preserve local distances: in other words a family of functions $f \in S^2 \rightarrow K, K \subseteq \mathbb R^2$ such that for ...
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1answer
25 views

Finding $y$ In Calculus(Area) Problem? [duplicate]

Find the number b such that the line $y=b$ divides the region bounded by the curves $y = x^2$ and $y = 4$ into two regions with equal area.
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3answers
57 views

How to solve this differential equation please?

I'm trying to solve: $$\frac{dz}{dx}+2xz=2x$$ I have got the integrating factor as $$e^{\int 2x dx}=e^{x^2}$$ and so $$ze^{x^2}=\int {2xe^{x^2}} dx+ C$$ But I don't know how to proceed it's mainly ...
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2answers
56 views

Finding the rate of rising water.

Water is pouring into a conical tank at a rate of 8 cubic feet per minute. If the height of the tank is 12ft, and the radius of its circular opening is 6ft, how fast is the water level rising when ...
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3answers
42 views

Show differentiability at a point then find differential where $f(x,y) = (x^2, xy+y^2)$

Show differentiability at a point then find differential where $f(x,y) = (x^2, xy+y^2)$ Want to show that $f(x,y) = (x^2, xy+y^2)$ is differentiable at $(a,b)$ and then calculate the differential ...
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1answer
14 views

By using the Chain Rule, find d$g(u_0,v_0)$ where $f(x,y) = x^2 + xy - y^2$ and $g(u,v) = f(u^2,uv)$

By using the Chain Rule, find d$g(u_0,v_0)$ where $f(x,y) = x^2 + xy - y^2$ and $g(u,v) = f(u^2,uv)$. $(x,y) \in \mathbb{R^2}$ and $(u,v) \in \mathbb{R^2}$ My Thoughts So I understand that $g(u,v) ...
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45 views

Questions about derivatives with e and ln in them

When taking the derivative of e^x I was under the impression it remained as e^x, but some books I've been reading have confused me.They show things like e^ -x and take the derivative as I mentioned ...
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23 views

is F differentiable at x0,y0

$F(x,y)=(x^2-y,xy)$, how to show that F is differentiable at $(x_0,y_0)$ and find $dF(x_0,y_0)$ My attempt: I think I understand part 2 of this question, $dF=\begin{pmatrix} 2x & -1 \\ y & x ...
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1answer
38 views

first order approximation of scalar function of matrix ( Mahalanobis distance)

I have tried to compute the 1st order approximation using Taylor's expansion of the Mahalanobis distance: $f(\mathbf{X})=\mathbf{a^TXa}$, where $\mathbf{a}\in \mathbb{R}^N$. The function maps ...
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12 views

Polylogarithm and unclear statement

I am trying to solve this question which may not have an answer at all, but any clarification would be much appreciated. I also tried to explain what I have tried/thought about it below. Let ...
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1answer
14 views

“Symmetric” numerical computation of second derivative

When numerically computing a first derivative, it is better to use $$f'(x) \approx \frac{f(x + \Delta x / 2) - f(x - \Delta x / 2)}{\Delta x}$$ than to use $$f'(x) \approx \frac{f(x + \Delta x) - ...
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1answer
20 views

Implicit Differentiation Solution Verification [Inverse Trig Function]

Find $\frac{dy}{dx}$. $$ \\ \\ \text{ } \\ \arctan{y^3} = \sin^3{x} + \cos^3{(yx)} \\ \text{ } \\ y^3 = \tan{(\sin^3{x} + \cos^3{(yx)})} \\ \text{ } \\ \frac{d}{dx}[y^3] = ...
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1answer
42 views

Proving inequality

Let $f$ be a twice differentiable function and let M, N, and P be the least upper bounds of |$f$(x)| |$f'$(x)| and |$f''$(x)| respectively prove that the square of N can never exceed 4 MP. I thought ...