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

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-3
votes
1answer
43 views

Use the definition of derivative for $f(x)=\begin{cases}x^2\sin\frac{1}{x}, \;x\ne 0\\0, \; x=0\end{cases}$

$f(x)=\begin{cases}x^2\sin\frac{1}{x}, \;x\ne 0\\0, \; x=0\end{cases}$ The best I can do so far is getting it into the following form: $\lim \limits_{x \to c}\frac{x^2\sin\left(\frac{1}{x}\right)-c^...
0
votes
2answers
31 views

The greatest value of $\cos(xe^{\lfloor x \rfloor} +7x^2 -3x)$

The greatest value of $g(x)=\cos(xe^{\lfloor x \rfloor} +7x^2 -3x)$ , $x\in [-1,\infty)$ , is My work: For function to be maximum $f(x) = xe^{\lfloor x \rfloor} +7x^2 -3x$ must be minimum When $ x ...
3
votes
2answers
96 views

Show differentiability of $f(x) = \sqrt{x^4 + y^4}$ in $(0,0)$

I'm trying to show the differentiability of $$f:\mathbb R^2 \to \mathbb R\text;\quad f(x) = \sqrt{x^4 + y^4}$$ in (0,0). Here's my attempt: Since $\partial_xf(x,y) = \frac{4x^3}{2\sqrt{x^4+y^4}}$ we ...
0
votes
2answers
37 views

Antiderivative for a function for integration

I have: $$f(x)=\cos(x) \times e^{\sin(x)}$$ and the fitting Antiderivative: $$F(x)=e^{\sin(x)}$$ Can someone please explain to me how I get from $f(x)$ to $F(x)$? In small steps? Thanks!
1
vote
1answer
28 views

Manipulation of Polynomials and Goodness-of-Fit

This question is directly connected with my problem in Python section HERE. Basically I've programmed a method with mathematical and physical background that forms a polynomial of selected degree ...
0
votes
1answer
33 views

derivative logical meaning basic question

The question is about the logical meaning of $\frac{df(x-1)}{dx}$, I did a single transform, which is $\frac{df(x-1)}{dx} = \frac{df(x-1)}{d(x-1)} * \frac{d(x-1)}{dx}$, since latter part $\frac{d(x-1)}...
1
vote
1answer
54 views

Differentiable functions satisfying a certain property

Whilst thinking about this question, I came across a problem. The original question was for what differentiable functions $f:\mathbb{R} \to \mathbb{R}$, with $y=f(x)$, does $\frac{dy}{dx} = F(y)$ for ...
2
votes
0answers
66 views

Calculus, continuity of the derivative at point [closed]

Let $f:I \to \mathbb{R}$ differentiable in the interval $I$ and let $a\in I$. For any sequences $(x_n)_{n\in \mathbb{N}}$ and $(y_n)_{n\in \mathbb{N}}$ in $I$ such that $\lim x_n = \lim y_n = a$, $x_n ...
0
votes
1answer
63 views

How to find $f(x)$ given $f'(x)$

I am dealing with a problem where I have the derivative $f'(x)$ in function of its antiderivative $f(x)$. How can I solve this? $f'(f(x))$ is linear (just assume any linear function, for sake of ...
2
votes
1answer
59 views

When do differential operators commute?

Given that the equation of motion of a particle placed on the apex of Norton's Dome is $$\frac{d^2 r}{dt^2}=r^{1/2}\qquad\longleftarrow\text{as proved in this previous question}\tag{1}$$ Solve this ...
1
vote
4answers
41 views

Find the number of roots lie in interval

Let $a \in R $ and let $f : R \rightarrow R $ be given by $f(x)=x^5 -5x + a $ Then $f(x)$ has three real roots if $a \gt 4$ $f(x)$ has only one real roots if $a \gt 4$ $f(x)$ has ...
2
votes
6answers
295 views

Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$

Can anyone give a rigorous proof of the derivative of this type of function? Specifically showing, $\frac{d(x^w)}{dx} = wx^{w-1}$ for a real $w$? I tried to use the Taylor series expansion for $(x+...
1
vote
1answer
45 views

Sinc function derivative formula

I was trying to find a formula for the derivative of the following function $$ f_{\alpha}(x) = \frac{\sin(\alpha x)}{x} $$ Since $$ \sin^{(k)}(\alpha x) = \alpha^k g_k(\alpha x), $$ where $$ g_k(\...
1
vote
2answers
43 views

Prove or Disprove that $f(x)$ cannot be $C_3$

Consider a function $y=f(x)$ with a single argument x with the real number line as its domain. Fix a real number $Q>0$, and suppose the following: $f(x)=f'(x)=f''(x)=0$ for all x from the interval ...
0
votes
3answers
37 views

How to show this function is differentiable everywhere?

Let $f: \mathbb{R} \rightarrow \mathbb{R}: x \mapsto x |x|.$ I plotted this function with Wolfram, and I see it is smooth everywhere so I figure it must be everywhere differentiable. I wanted to ...
3
votes
2answers
49 views

Show that the equation of motion for a particle on Norton's Dome is $\frac{d^2 r}{dt^2}=r^{1/2}$

A particle sits at the top of a dome, whose height drops away from the centre, with a drop given by $$h=\frac{2r^{3/2}}{3g}$$ where $g$ is the acceleration due to gravity, and $r$ is a coordinate ...
0
votes
1answer
24 views

Modulus in Calculus?

Does Modulus function has any effect during differentiation and integration of a quantity? For example: Let two velocities be: $$ v_1= (t-2) m/s $$ and $$ v_2=|t-2|m/s $$ If we differentiate them ...
2
votes
1answer
99 views

Proving a function is not differentiable, when its partials are not continuous

Let $$f(x,y)=\frac{y \sin (3 x)}{\sqrt{x^2+y^2}},$$ and $f(0,0)=0$. I'm trying to prove that it's not differentiable in $(0,0)$. Some my plan was to compute the limit of the definition of ...
1
vote
1answer
34 views

Derivative for function at particular value

How to find $\dfrac{\text{d}y}{\text{d}x}$ at $x=\dfrac{\pi}{4}$ for: $$y=(\log_{\cos}\sin(x))(\log_{\sin}\cos(x))^{-1}+\sin^{-1}\left(\frac{2x}{1+x^2}\right)$$
1
vote
6answers
61 views

Solving second order differential equation 3

How do one solve this? $$\begin{align}\frac{d^2y}{dt^2}+e\frac{dy}{dt}+1=0,&&y(0)=0,&&\frac{dy(0)}{dt}=1\end{align}$$ The exact solution of above equation is $$y(t)=\frac{1+e}{e^2}(1-...
0
votes
1answer
35 views

Slope of this tangent

Okay I have to find the slope of this tangent to the curve $y=\int_0^x \frac{dx}{1+x^3}$ at the point where $x=1$. My try- I integrated the expression and differentiated it afterwards to get the ...
2
votes
0answers
41 views

Let $f$ be a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(x)f''(x)\leq 0$ for all $x\in \mathbb{R}$. [duplicate]

Let $f$ be a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the derivative $f'(x)$ and all its successive derivatives exist everywhere and also $f(x)f''(x)\leq 0$ for all $x\in \mathbb{R}$. ...
0
votes
0answers
43 views

Find the derivative of $y = \frac{\sqrt[3]{4x^3+8}}{(x+2)^5}$ [duplicate]

Can anyone solve 1b? Find the derivatives of each of these functions. b. $y = \frac{\sqrt[3]{4x^3+8}}{(x+2)^5}$ I am really confused, I used the quotient rule for differentiation of the ...
0
votes
4answers
84 views

Find the derivative of $y= \frac{(4x^3 +8)^{\frac{1}{3}}}{(x+2)^5}$ [closed]

How can we find the derivative of $y= \frac{(4x^3 +8)^{\frac{1}{3}}}{(x+2)^5}$? so far this is what I have done, and am confused about what to do after? sorry for the messy handwriting
1
vote
3answers
142 views

Find range of $f(x)=3^x+5^x-8^x$

Find range of $f(x)=3^x+5^x-8^x$. My attempt: On observation one sees that $f(1)=0$. On taking $g(x)=\left(\frac{3}{8}\right)^x+\left(\frac{5}{8}\right)^x-1$ and then observing that $g'(x)=\left(\...
3
votes
3answers
84 views

Application of Derivatives rigorous proof

Let $f:R\rightarrow R$ be a function such that all its successive derivatives exist in all $R$ and also $f(x)f''(x)\leq 0$ everywhere. If $\alpha$ and $\beta$ be two successive roots of $f(x)=0$. ...
3
votes
3answers
71 views

$x(a^{1/x}-1)$ is decreasing

Prove that $f(x)=x(a^{1/x}-1)$ is decreasing on the positive $x$ axis for $a\geq 0$. My Try: I wanted to prove the first derivative is negative. $\displaystyle f'(x)=-\frac{1}{x}a^{1/x}\ln a+a^{1/...
0
votes
4answers
58 views

If $y=x^2$ then $\frac{dy}{dx}=2x$, which means if we increase $x$ by 1, $y$ would change by $2x$, but it actually follows $2x+1$, why?

Say $x=5$, then $y=25$ and $\frac{dy}{dx} = 10$.But $6^2-5^2 = 11$ What am I doing wrong?
0
votes
3answers
46 views

Differentiability of function with respect to its continuity

Let $f:\mathbb{R}\rightarrow\mathbb{R}$, with $$ f(x)=\begin{cases} \dfrac{1-e^{-x}}{x}, & x<0\\[4px] a, & x=0\\[6px] \dfrac{\ln(1+x)}{x}, & x>0 \end{cases} $$ where $a\in\mathbb{R}$...
3
votes
2answers
95 views

Derivative of $X_u A X B X_u^T$ w.r.t. $X_u$

How to solve this $\frac{d X_u A X B X_u^T}{d X_u}$, where $X, A, B \in \mathbb{R}^{n \times n}$ and $X_u$ is the $u$-th row in $X$?
5
votes
7answers
117 views

How can a $\sin x$ come out of the equation $\frac{d^2}{dx^2}f(x)=-f(x)$ as the solution, while there's no sign of a trigonometric function in it?

This is a differential equation: $$\frac{d^2}{dx^2}f(x)=-f(x)$$ Turns out that the answer is $\sin x$. But HOW?! It is impossible to achieve a trigonometric function by integrating that equation. ...
5
votes
4answers
147 views

Why is this proof of the chain rule incorrect?

I saw this proof of the chain rule but it says this is a flawed proof. Why? I guessed the reason it is wrong because you can't substitute $g(x+h)$ and $g(x)$ into in limit.
1
vote
1answer
32 views

Derivative of an integral on a set-intersection

If $f$ is a real-valued integrable function, $a\in \mathbb{R}$ is some constant, and $$F(x):= \int_{t\in [a,x]}f(t)dt,$$ then: $$F'(x) = f(x)$$ What if, for some constant set $B\subseteq \mathbb{R}$: ...
4
votes
4answers
83 views

Showing that $tf(x) + (1-t)f(y) \leq f(tx + (1-t)y)$

Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ is a twice continuously differentiable function such that $f''(x) \leq 0$. Prove that $$tf(x) + (1-t)f(y) \leq f(tx + (1-t)y)$$ for any two points ...
3
votes
4answers
62 views

Use the chain rule to evaluate $\frac{\mathrm{d}}{\mathrm{d}x}\displaystyle\int_{x^2-1}^{\sin(x)} \cos(t) \, \mathrm{d}t $

Doesn't the derivative of that integral just equal $\cos(x)$? What does it mean to use the Chain Rule? I know for sure it has nothing to do with $u$-substitution. Any help would be appreciated, thanks....
1
vote
1answer
55 views

Using higher order derivatives

I am currently learning about the general Notion of Differentiability. I came across some difficulties when working with higher order derivatives and I am hoping for confirmation or comments on some ...
1
vote
2answers
47 views

Functions of Several Variables - Successive Differentiation

I am struggling with this question: Regarding u and v as functions of x and y and defined by the equations : $$ x=e^u \cos(v) $$ $$ y=e^u \sin(v) $$ show that : $$\frac{\partial^2z}{\partial x^...
1
vote
1answer
31 views

Property of the covariant derivative

I am learning to use the covariant derivative. In particular, I am trying to verify the expression $${\nabla}_{b}[(\nabla^a S) (\nabla_a S)] = 2 \nabla^a S \nabla_b \nabla_a S$$ for an arbitrary ...
1
vote
1answer
72 views

Is a weakly differentiable function differentiable almost everywhere?

I am working with Sobolev spaces. Let's suppose $\Omega \subset \mathbb{R}^n$ is an open set. A function $u: \mathbb{R}^n \to \mathbb{R}$ in $L^1(\Omega)$ is said to be weakly differentiable if there ...
6
votes
1answer
91 views

Differentiation under the integral sign for volumes in higher dimensions

Consider a smooth convex/compact domain $D\subset \mathbb{R}^n$ and a smooth, concave function $F:D\to \mathbb{R}$. Then we can define the function that simply takes the volume of the upper contour ...
1
vote
0answers
35 views

Gradient of a function involving maxima

How do I find the gradient of a function like $f(\vec{v})$ where $$ f(\vec{v}) = \max_{\vec{t}\geq 0} g(\vec{v},\vec{t})$$ For example, I have a function defined as follows: \begin{align} f(\vec{v}) ...
0
votes
1answer
15 views

General guidelines to solve Mean Value Theorem problems

I am wondering if there is a general guideline to solve this specific type of MVT problem. For the teachers: how do you explain to the students how to apply MVT for these two questions below? ...
1
vote
1answer
33 views

Derivative of function defined as Integral

I have to find all partial derivatives of: $$ f(x,y,z) = \int_{\cos x + \sin y}^{z} e^{tz} dt $$ I easily get confused with all this variables, but the idea is to use The Fundamental Theorem of ...
3
votes
2answers
75 views

What does it mean when the second derivative at a point is infinite?

What does it mean when the second derivative of a function at a certain point is infinite? What would be neat examples that illustrate what happens? When the first derivative is infinite you get a ...
1
vote
5answers
67 views

When is this function increasing?

The function is given by $f(x)=\sin^{4}x+\cos^{4}x$. My try -- Okay so I differentiated this function and if I solved correctly which I hope I did I got $ \sin2x(\cos2x)>0 $.But im not getting the ...
-7
votes
3answers
247 views

Where is $f(x)=x+\frac{1}{x}$ decreasing?

Given a function $f(x)=x+\frac{1}{x}$ How do I find the interval in which it is decreasing? My try: I differentiated the function using the division rule and finally got $f'(x)=x^{2}-1$. How do I ...
0
votes
3answers
64 views

Finding $n$th derivative in an unusual way

If $f(z) = \frac{e^{iz}}{z^2-1}$, then $f^{(4)}(z)$ can be found by differentiating $f(z)$ four times. I tried to use Cauchy's integral formula, but the integrand is not holomorphic at $z=0$, so we ...
0
votes
3answers
28 views

Nature of function(increasing/decreasing)

Okay so I just wanted to ask the nature of this function $f(x)=\frac{e^{2x}-1}{e^{2x}+1}$ that is ;whether it will be decreasing or increasing. $$ $$ I know that if we diffrentiate a function with ...
0
votes
2answers
33 views

Find $f'(c)$ using the derivative definition for $f(x)=\frac{1}{x^2}$

So $f:\mathbb R \backslash\{0\}\rightarrow\mathbb R$ by $f(x)=\frac{1}{x^2}$ I'm need to use the definition of the derivative to find $f'(c)$ for $\frac{1}{x^2}$. If I use standard differentiation ...
0
votes
1answer
33 views

derivative transpose

I'm reading the book "The Elements of Statistical Learning - Data Mining, Inference, and Prediction" chapter 3 and there comes a simple derivation that I don't understand: We have: $...