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

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3
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2answers
54 views

If $s(x)= \int \sqrt{1+ \Big(\frac{dy}{dx}\Big)^2} dx$, what is $x(s)$?

$s(x)$ is the formula for arc length of a function $f(x)=y$. In the book I'm studying, curvature is defined as the instantaneous rate of change of direction (inclination of angle $\theta$) with ...
1
vote
2answers
45 views

Use L'Hopital's rule to show that u′(t) = −λ u(t) (question from a Probability course)

I'm currently in a probability course and my professor suggested this question: Consider a continuous and differentiable function u(t) such that u(t + s) = u(t)u(s), for any s ∈ R and t ∈ R. This ...
0
votes
1answer
19 views

Df(x) of $f:\mathbb R^{n^2}\rightarrow\mathbb R^{n^2}$

I am a bit a confused about finding $Df(x)$ of functions like $f:\mathbb R^{n^2}\rightarrow\mathbb R^{n^2}$. For example the derivative of the function $f:M_n(\mathbb R) \rightarrow Sym_n(\mathbb R)$ ...
0
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3answers
35 views

Given Position equation Solve for t

This is calculus, and here is given information: Position of a spaceship at t hours after leaving planet is $$s(t)=.004t^3 + 400t$$ in thousands of miles. The distance between planet and destination ...
5
votes
3answers
463 views

On the exactness of the calculus formulas

Are calculus formulas, differentiation and integration, exact formulas or are some approximations involved? That is, is the value of a definite integral of a function the exact value of the (signed)...
2
votes
3answers
45 views

Did I derive this correctly? More questions inside

Sorry for the title, didn't quite know what to title it since I had a few questions. Anywho, I'm working on my homework (finding derivatives of the given function) and had a few questions. $y=\ln\...
2
votes
0answers
40 views

Derivative of an eigenvalue with respect to tensor itself

I have a rank-2 tensor $\mathbf{C}=C_{\alpha\beta} \mathbf{A}^\alpha \otimes \mathbf{A}^\beta$ defined in a curvilinear coordinate system. I want to compute the derivative of an eigenvalue $\Lambda_i$ ...
0
votes
1answer
37 views

If $\Delta$ is the area of the triangle formed by the positive x axis and the normal and tangent …

Problem : If $\Delta$ is the area of the triangle formed by the positive x axis and the normal and tangent to the circle $x^2+y^2=4$ at (1,$\sqrt{3})$ then find the value of $\Delta$ My approach : ...
0
votes
2answers
28 views

Solve for $\frac{dy}{dx}$ of a trigonometric function after implicit differentiation

I'm supposed to implicitly differentiate $\sin(x+y)=2x-2y$. I've already taken the first derivative and got $$ \left(\frac{dy}{dx}+1\right)\cdot\cos(y+x)=-2\left(\frac{dy}{dx}-1\right) $$ www....
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vote
2answers
54 views

Geometric solution to non-linear differential equation

Let the non-linear differential equation be $\dot{x}= sin\left ( x \right )$ Plotting this on the graph, we have the standard sin curve with the exception that the horizontal axis is labelled $x$ and ...
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0answers
8 views

Graph sketching trigonometric functions

How would I go about graphing sin(sin(x)) and cos(cos(x))? Without use of a calculator I have shown they are never equal to each other. The only way I can go about it is to plug in values and ...
1
vote
3answers
70 views

How to solve ODE

Solve the DE: $$2y^2y''+2y(y')^2=1$$ Is it possible to solve this by implicit substitution i.e. let $v = y'$ and thus $$\frac{dv}{dy}v = \frac{1-2yv^2}{2y^2}$$ by the chain rule. And then from ...
0
votes
2answers
50 views

simplest possible way to prove chain rule for second derivative

Given $\phi=\phi(u)$ and $u=u(x)$ What is the simplest possible way to prove $$ \frac{d^2\phi}{dx^2}=\bigg(\frac{du}{dx}\bigg)^2.\frac{d^2\phi}{du^2}+\frac{d^2u}{dx^2}.\frac{d\phi}{du} $$
0
votes
1answer
64 views

Finding derivatives , where am I going wrong?

What I've done: $$\frac{df}{dx} = 15y^3 \cos(3x)- 10xy^6 +2e^x \sin(y)$$ $$\frac{d^2f}{dx^2} = -45 y^3 \sin(3x) - 10y^6 + 2e^x \sin(y)$$ $$\frac{df}{dy} = 15 y^3 cos (3x) - 30 x^2 y^5 +2 e^x \cos(y)...
3
votes
4answers
72 views

Finding $d^2y/dx^2$

A while ago I did a problem where I needed to find $\frac{d^2y}{dx^2}$ with $x=5+t^2$ and $y=t^2+t^3$. I found $dy/dx$ to be $1+\frac{3t}{2}$. But I never wrote down how I found $\frac{d^2y}{dx^2}$. ...
0
votes
0answers
61 views

Confusing: derivatives of a function w.r.t. each entry and each row vector of a matrix

It is a basic matrix factorization problem. Suppose I have a matrix Y and it was factorized into low rank matrices W and H (rank is $d$, where $d$ <= $n$ ). The following is the noation: $C \in \...
0
votes
3answers
31 views

What is the meaning behind the linear differential equation standard form?

In differential equations this form is often used to describe a differential equation: I'm confused what this equation is saying. If you add all the derivatives of a function of $x$ together you ...
0
votes
1answer
38 views

Higher Dimensional Derivatives

I've seen the definition for higher dimension derivatives, like the derivative of scalars with respect to vectors, and vectors with respect to scalars, and scalars with respect to matrices, but I'm ...
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3answers
53 views

Multivariable calculus/ How to solve problems involving implicit definition of a function in a function?

Hello all math geniuses out there here is a question that probably does require much thinking, but i can't seem to figure out. The equation $x^3 +yz-xz^2=0$ implicitly defines a function $z=f(x,y)$ ...
2
votes
3answers
33 views

Show that the equation of the tangent to the parabola $y^2=4ax$ at the point (p,q) is $qy=2a(x+p)$

Question: Show that the equation of the tangent to the parabola $y^2=4ax$ at the point (p,q) is $qy=2a(x+p)$ These are my two approaches: First approach: If we have $(p,q)$ as $(x_1,y_1)$ $$y^...
0
votes
3answers
114 views

Why do we not consider infinity in calculus?

We know that $$\frac{d}{dx}(\log x)=\frac{1}{x}$$ Conversely, $\int \frac{1}{x} dx$ gives $\log x + c$. However, if we take $\frac{1}{x}$ as $x^{-1}$, from the formula $$\int x^n dx = \frac{x^{n+1}}{...
1
vote
1answer
30 views

Derivative of $x(x+2)^3$ in factored form don't see it

Studying for a big test trying to do the derivative of $x(x+2)^3$. I did the product rule and got $3x(x+2)^2 + (x+2)^3$. Pretty standard stuff. I got half credit and was told to simplify it like ...
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votes
2answers
1k views

Is calculating the summation of derivatives “mathematically sound”?

I have just discovered that if you take the following series: $$1 + x + x^2 + x^3 + x^4 + \cdot \cdot \cdot = \sum_{n = 0}^\infty x^n$$ and replace each term in the series with the derivative of them, ...
0
votes
1answer
34 views

Accuracy of derivatives as instant variation predictor

For the following equation: $f(x)=4x^2 + 16x - 10$ At x=1, y = 10 At x=2, y = 38 Given the derivative is $f'(x)=8x + 16$ and ...
4
votes
2answers
69 views

Swapping limits: $\lim_{h\to 0}\lim_{n\to \infty}\frac {(1+1/n)^{hn}-1}{h}=\lim_{n\to\infty}\lim_{h\to 0}\frac {(1+1/n)^{hn}-1}{h}$

Almost a year ago I asked the question: How to differentiate $e^x$? And in the accepted answer, the following equality appeared: $$\lim_{h\to 0}\lim_{n\to \infty}\frac {(1+1/n)^{hn}-1}{h}=\lim_{n\to\...
2
votes
0answers
47 views

Derivation of tetration by iteration

I was screwing around a bit differentiating tetrations and was trying to write some rules for them. I came up with this recursive definition: $$ \frac{d\ ^nt}{dt} =\ ^nt \cdot log(t)\frac{d\ ^{n-1}t}{...
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vote
0answers
51 views

Does there exist a derivative of the Golden Ratio (equation)?

On my calculus exam, there was a question asking "Is the Golden Ratio differentiable? If so, use the definition of a derivative to show it." The "definition of a derivative" is $f'(x)=\displaystyle\...
0
votes
0answers
32 views

How many solutions $f(x) = 6 \cdot \ln(x^2 + 1) - e^{x}$

For this function how many solutions there are? $$f(x) = 6 \cdot \ln(x^2 + 1) - e^{x}$$ I know how to find derivative $$f'(x) = \frac{12 x}{x^2 + 1} - e^{x}$$ but now what?
0
votes
2answers
52 views

How many solutions: $ a^{x} = x $ [closed]

I'm clueless. For $ a > 0 $ how many solutions: $ a^{x} = x $
3
votes
2answers
91 views

$n$-th derivative of $\log(1+x)/x$

What is the $n$-th derivative of $$\frac{\log(1+x)}{x}$$ Now I have seen some analytical methods of getting $n$-th derivative of nicer looking functions such as the $n$-th derivative of $$ 1\over 1+...
0
votes
1answer
21 views

Differentiating a Polynomial in brackets.

I have the question below: My question here is as you can see I have dropped the power of two down in-front of the bracket and reduced the power. Is this way of solving the question incorrect? I have ...
2
votes
1answer
72 views

Complex continuity and differentiability of a piecewise function

Let $g(z):\mathbb{C}\rightarrow\mathbb{C}$ with $$g(z)=g(x+iy) = \begin{cases} \dfrac{x^2y+ixy^2}{x^2+y^2} & : (x,y)\neq (0,0), \\ 0 & : (x,y)=(0,0). \end{cases} $$...
9
votes
3answers
251 views

100th derivative of $e^{-x^2}$ at point $0$

Problem: Find $\frac{\mathrm d^{100}}{\mathrm dx^{100}}e^{-x^2}$ at point $0$. My attempt: $y'=-2xe^{-x^2}$ I tried to use General Leibniz rule and I didn't get much better information. Without: ...
0
votes
3answers
65 views

$nth$ derivative of log function

The question is , what is nth derivative of $\log(4-x^2)$? I know how to solve general derivatives, but after I encountered with this I am completely bewildered.
0
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1answer
29 views

Understanding of Example of Theorem

Theorem Suppose $\,f\left( a,b\right) \rightarrow \mathbb{R}\,$ is differentiable. Then $f'$ satisfies the intermediate value theorem. Example $$f\left( x\right) =\begin{cases} 1,x\leq 0\\ 0,x>0\...
1
vote
1answer
51 views

How to differentiate $y = C(x)e^{-\int p(x) dx }$

In a book following equation: $$ y = C(x)e^{-\int p(x) dx } $$ is differentiated into: $$ y' = C'(x)e^{-\int p(x) dx } + C(x)\left( -p(x) \cdot e^{-\int p(x) dx }\right ) $$ but there are no ...
1
vote
1answer
45 views

How do I solve this Lagrange multiplier question?

A function is defined by $f(x,y)=x^4 - 6x^2y^2 + y^4 -2x^2 + 2y^2$. If we let $z=f(x,y)$ and let a particle travel in the direction which $f$ decreases most rapidly, how do I show that $xy(x^2 - y^2 -...
10
votes
4answers
311 views

Finding Maximum Area of a Rectangle in an Ellipse [duplicate]

Question: A rectangle and an ellipse are both centred at $(0,0)$. The vertices of the rectangle are concurrent with the ellipse as shown Prove that the maximum possible area of the ...
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vote
1answer
34 views

On differentiable functions on real line satisfying $f'(x)\ge f(x)^2 , \forall x>0$

Does there exist a real valued differentiable function $f$ on real line such that $f'(x) \ge f(x)^2 , \forall x >0$ ? If such a function exist , must it be twice differentiable or at least ...
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vote
2answers
44 views

Differentiability of $\int_0^tx^2f(x)dt$

If $f(x)$ is continuous, how can I prove that $\int_0^tx^2f(x)dt$ is differentiable? This is what I thought of: Since $\int_0^tx^2f(x)dt=F(t)-F(0)$ for some function $F$ which is the antiderivative ...
4
votes
4answers
93 views

Working with the $\frac{d}{dx}$ operator

I have a fundamental query about the way derivatives can be used in algebraic manipulations. Say $\dfrac{d(\ln x)}{dx}=\dfrac{1}{x}$ Apparently, this can be manipulated to $d(\ln x)=\dfrac{dx}{x}$. ...
0
votes
2answers
24 views

Differentiation for term in Parenthesis

What is the derivative of $(4 - 9x^4)^{\frac{1}2}$? My answer is $\frac{1}2(4 - 9x^4)^{-\frac{1}2}$ But the answer is $-18x^3(4-9x^4)^{-\frac{1}2}$ Why is my answer not correct?
0
votes
1answer
32 views

Partial derivative of $\frac{d}{dx}(x^2-2sin(y))$

WolframAlpha computes $\frac{d}{dx}\left(x^2-2(\sin y)\right)$ to be $$ \frac{d}{dx}\left(x^2-2 (\sin y)\right) =2 \left(x-(\cos y)y'(x)\right) $$ entered "differentiate (x^2 - 2*sin(y)) with ...
3
votes
1answer
101 views

If $\,f''(x) \ge f(x)$, for all $x\in[0,\infty),$ and $\,f(0)=f'(0)=1$, then is $\,f(x)>0$?

Let $f:[0,\infty) \to \mathbb R$ be a twice differentiable function, such that $\,f''(x) \ge f(x)$, for all $x\in\ [0,\infty)$, and $ f(0)=f'(0)=1$. Can we deduce that $f$ is increasing? I feel ...
0
votes
4answers
76 views

Differentiate $y=x + \frac{1}{x+\frac{1}{x+\frac{1}{x+…}}}$ [closed]

If $y=x + \dfrac{1}{x+\dfrac{1}{x+\dfrac{1}{x+...}}}$ then $\dfrac{dy}{dx}$ equal to ?
1
vote
1answer
45 views

Can we have a continuous choice in the mean value theorem

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be differentiable. Must there exist a continuous function $g:\{(a,b)\in \mathbb{R}^2: a<b\}\rightarrow \mathbb{R}$ such that: For every two distinct real ...
0
votes
0answers
33 views

Taking the zero derivative

I am trying to find the derivatives of a simple function but in increasing order, i.e. zero, first, second and so on. However, I am unsure on what does the 'zero' derivative gives. For example, if I ...
1
vote
1answer
40 views

Show $f$ is differentiable at the endpoint with $a$ with $f'(a)=c$

Let $f: [a,b] \rightarrow \mathbb{R}$ continous on [a,b] and differentiable on $(a,b)$. Asume there exist some $c \in \mathbb{R}$ so $f'(x) \rightarrow c$ for $x \rightarrow a+$. Show $f$ is ...
1
vote
1answer
36 views

How to do matrix derivative calculation?

For example, the derivative $W^{T}AW$ with respect to $W$ is $2AW$. Is there any general guidelines or rules that we can follow?
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0answers
69 views

If $a+b+c = 0$ then the quadratic equation $3ax^{2}+2bx +c=0$ has atleast one root in _________?

If $a+b+c = 0$ then the quadratic equation $3ax^{2}+2bx +c=0$ has atleast one root in _________? Rolle's theorem states that if $f(a) = f(b)$ then there exists a $p \in [a,b]$ such that : $f'(p) = \...