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

learn more… | top users | synonyms (2)

1
vote
4answers
14 views

Differentiability question ends up in contradiction.

Let $f(x)=x^3cos\frac{1}{x}$ when $x\neq0$ and $f(0)=0$. Is $f(x)$ differentiable at $x=0$? My first attempt Definition: A function is differentiable at $a$ if $f'(a)$ exists. $$f'(x)=\lim_{h ...
0
votes
2answers
23 views

When to use the chain rule

Would I use the chain rule in the following derivative problem: $$(sinx/x)$$ So far I have simplified it to: sin(x)(-1x^(-2))+x^(-1)(cosx) Would I have to further take the derivative of cosx ...
2
votes
1answer
63 views

Differentiate $e^x+x^e$

My answer was $e^x+ex^{e-1}$. As I understand it, the derivative of $e^x $ is $e^x$. As for $x^e$, I made this $e(x^{e-1})$ and simplified from there. Where was my error?
0
votes
1answer
31 views

Related rates: Rocket launch. dy/dt = 0.8, y = 5, x =?

There is a vertical rocketlaunch. x represents horizontal distance y represents vertical distance Right after launch: y = 5 ; dy/dt = 0.8 Given: Distance between launch pad and radarstation: x1 = ...
0
votes
0answers
23 views

differentiation of log of a sum of vectors

I would very much like to be able to differentiate the following function with respect to $x$: $$ln(\sum \limits_{j=1}^{d} \theta_j \bf a_j^2)$$ Where $\bf a_j$ is a $d \times 1$ vector with $x$ in ...
1
vote
0answers
43 views

derivative of normalizer in exponential form — change integral and gradient

When deriving the relation between normalizer and expectation of the sufficient statistic for distributions in exponential form one uses the fact, that the density integrates to one: $$1 = ...
0
votes
1answer
33 views

Use Leibniz' formula to show that the $(2n)$th derivative of $(2x^2 + 3x +1)sinx$ is $(-1)^n(2x^2+3x-8n^2+4n+1)sinx+(-1)^{n+1}(8nx+6n)cosx$ wrt $x$

If I let $f=f(x)=sinx$ and $g=g(x)=2x^2+3x+1$ and $D=$ First derivative wrt $x$, $D^2=$ Second derivative wrt $x$ and $D^n=$ $nth$ derivative wrt $x$ then, Leibniz' formula states that $\displaystyle ...
2
votes
1answer
26 views

Total Time Taken Question

Distance of chord = Time taken to "swim" to the desalination plant = I'm stuck here! The textbook working out is as such: I don't understand how they have the 'k' or 1/2 the runs river at ...
0
votes
1answer
23 views

Shortest Chord from origin to function

worked solution: Is this found using the distance of a line equation, where instead of co-ordinate points they use functions, so the two functions are g(x) and x (because the origin is on the ...
1
vote
1answer
18 views

Finding implicit differentiation using power rule.

Original Function: $(\sin(\pi x)+\cos(\pi y))^2 = 2$ Step1: $2(\pi \cos(\pi x) - \pi \sin(\pi y)\frac{dy}{dx})(\sin(\pi x)+\cos(\pi y))$ Step2: $(2\pi \cos(\pi x)-2\pi \sin(\pi ...
0
votes
2answers
67 views

Integrate $\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$ [closed]

Evaluate$$\int_{0}^{\pi} \frac{1}{a-b\cdot cos(x)}$$ Solution through either contour integral method or indefinite integral method please!
2
votes
3answers
99 views

Problem with roots

I am having a few problems with roots. This is apart of a larger question where I am taking the derivative of of a function. I know I got the first part right (answer key) but when I plug in root 2 ...
0
votes
3answers
26 views

Using product rule or chain rule

For the equation $y=e^xx-2e^x$, I used product rule for the left ($e^xx+e^x)$ and right ($2e^x$) and combined the two to get $y'=e^xx-e^x$. But when I tried to factor out $e^x$ in the original ...
2
votes
1answer
34 views

Finding the equation of a line perpendicular to a curve at a given point.

Find the equation of the perpendicular line of $y=e^{-2x^2}$ at the point where $x=1$. I found the derivative: $y'=e^{-x^2}-2x$. And when I plug in one to the derivative I get: $m=\frac{1}{e}-2$. I ...
0
votes
3answers
50 views

What is the difference between a Limit and Derivative?

When you get the derivative of a point, arn't you just getting the limit at that point? I'm not quite sure why they need to be named differently when they seem to be doing the same thing.
0
votes
1answer
15 views

Find the domain of derivative of the function $f(x)=\mid \sin^{-1}(2x^2-1)\mid$.

Find the domain of derivative of the function $f(x)=\mid \sin^{-1}(2x^2-1)\mid$. I was a little confused about the modulus. I can do the derivative and even calculate the domain, but, the modulus ...
1
vote
1answer
48 views

Transform the following Differential Equation

Transform the differential equation: \begin{equation} x^2\frac{\partial{z}}{\partial{x}} + y^2\frac{\partial{z}}{\partial{y}}=0 \end{equation} Taking as new variables $u=x$, $v=1/y-1/x$, ...
4
votes
2answers
88 views

What intuition stands behind implicit differentiation

I'm trying to undestand implicit differentation Let's take as a an example equation y^2 + x^2 = 1 1. How i think about how the equation works I think the function as : if x changes then the y term ...
0
votes
1answer
34 views

Show that $f'_+(a)=f'(a+)$ if both quantities exist.

Show that $f'_+(a)=f'(a+)$ if both quantities exist. I'm not really sure where to start, any help is appreciated. I came up with this: If $f'(a^+)$ exists, then by definition $f'(a+) = \lim_{x\to ...
1
vote
2answers
48 views

Intermediate step for integration?

Among the various method of Integration there is one specific method(it may vary according to the terms) where for instance if we have a function as:$$\frac{px+q}{ax^2+bx+c}$$ To integrate this we ...
2
votes
3answers
40 views

Trigonometric Identity for tan?

Can somebody please help with this (probably simple) query. Given $$dx = R\,d(\tan\theta)$$ this can be expressed as $$dx = R\,\sec^2\theta\,d\theta$$ I can't determine where the ...
0
votes
1answer
49 views

Exam Question from multivariable calculus.

This question from a previous multivariable calculus exam.I don't know how to start with this question: Let $f$ be differentiable at every point of line segment joining $x_0$ and $x_0+h$.Show that ...
1
vote
1answer
59 views

A question on Lie derivative

For a Lie derivative $\mathscr{L}_{X} Y$ of $Y$ with respect to $X$, we mean that for two smooth vector fields $X$ and $Y$ on a smooth manifold $M$ such that the following holds $$ \mathscr{L}_{X} Y = ...
2
votes
0answers
34 views

Differentiation - a technical point

I understand the following equation to be correct, but why can we treat the differentials as fractions and cancel them out? What would be the correct way to view it? $$ \int_{-\pi/a} ...
0
votes
3answers
33 views

Is this derivative correct?

I'm newbie at Calculus, so I'm doing some exercises of derivates, I know by the formula: $f(x) = \sqrt u$ $\frac {df(x)}{dx} = \frac{u'}{2 \sqrt u}$ that the derivate of the next function is: ...
2
votes
1answer
20 views

If all directional derivatives are $0$, the function is constant.

Let $f:\mathbb{R}^{n} \rightarrow \mathbb{R}$ be differentiable in every point of the disc $ B_{r}(\vec{a})$. If $D_{\vec{y}}f(\vec{x})=0$ for $n$ linearly independent vectors $\vec{y}_{1}, ...
3
votes
1answer
48 views

A convex function is differentiable at all but countably many points

Let $f:\Bbb R\to\Bbb R$ be a convex function. Then $f$ is differentiable at all but countably many points. It is clear that a convex function can be non-differentiable at countably many points, ...
5
votes
2answers
70 views

nth derivative of $e^{-x}\sin(x)$

I'm trying but no luck. Can't find a pattern yet. The exercise is to find the nth derivative of $e^{-x}\sin(x)$ probably by induction.
1
vote
1answer
61 views

Proving convexity of a function whose Hessian is positive semidefinite over a convex set

C is a convex set in R^n and f:R^n --> R is twice continuously differentiable over C. The Hessian of f is positive semidefinite over C, and I want to show that f is therefore a convex function. I ...
0
votes
1answer
22 views

A Calculus Question, Derivatives

Please help and explain how to do the problem taking calc online and the teacher is not present right now
0
votes
1answer
36 views

Finding the Newton map

Start with $p(x)=(x-x_0)^k g(x)$. I need to find the Newton map, which is $Np(x)=x−p(x)/p'(x)$. Is $p'(x)=k(x-x_0)^{k-1}g(x)+(x-x_0)^kg'(x)$? I'm having a tough time with $k$ and $x_0$.
0
votes
1answer
38 views

Does calculating d/dt of someething mean the same as calculating the derivative?

Probably a dumb question but I missed college for a week due to sickness. The exercise I have to do is: d/dt eight root of t^7. Does this simply mean I have to calculate the derivative of the eight ...
0
votes
1answer
23 views

Differentiability of $0 \leq f(x,y) \leq |x|^\alpha \cdot |x|^\beta$

Let $f:\mathbb{R^2} \to \mathbb{R}$ such that $0 \leq f(x,y) \leq |x|^\alpha \cdot |x|^\beta$ $\forall{(x,y) \in B_{\delta_0}(0,0)}$ for some $\delta_0 > 0$. I have to prove that if ...
2
votes
1answer
103 views

Derivative of Softmax loss function

I am trying to wrap my head around backpropagation in a neural network with a softmax classifier, which uses the softmax function: \begin{equation} p_j = \frac{e^o_j}{\sum_k e^{o_k}} \end{equation} ...
1
vote
1answer
63 views

partial derivative of $f(x,y)$ who satisfies $f_{xx}-f_{yy}=0$

Suppose that $z=f(x,y)$ and its second-order partial derivative is continuous. It also satisfies $\displaystyle\frac{\partial^{2}f}{\partial x^{2}}-\frac{\partial^{2}f}{\partial y^{2}}=0$,$f(x,2x)=x$ ...
0
votes
1answer
37 views

Proving the given inequality using the concept of increasing-decreasing functions

I want to prove $x-(x^3/3)<\tan^{-1} x<x$ for all real $x>0$. I have done the following: If it could be proved that y has a maximum value of 0 at x=0 the first part of the given ...
0
votes
2answers
53 views

Derivative of mutual information

Here is the definition of mutual information $I(X;Y) = \int_Y \int_X p(x,y) \log{ \left(\frac{p(x,y)}{p(x)\,p(y)} \right) } \; dx \,dy,$ where $x$ and ...
0
votes
0answers
20 views

Derivatives and rates of change word problem

It's example 5; I understood everything up until the value of theta was found. I don't understand where the 25 ft came from! How was the length of the beam found to be 25 ft? So, yeah, that's the ...
1
vote
1answer
36 views

Help with dy/dx of natural log

Could someone explain how I would work this problem with steps?
2
votes
3answers
97 views

Is the function $y(t)$ is a solution of the equation $y'=\sin(yt)$?

Is the function $y(t)$ a solution of the equation $y'=\sin(yt)$? any thought to start me up? I'm not sure what is the question asking. EDIT: Someone tell me if I'm correct or not . If I'm finding ...
2
votes
3answers
168 views

Derivative as a linear transformation

i can't understand this : consider $f:A \rightarrow Y$ for normed space $Y$, $~~A$ is an open subset of $\mathbb R$ ,and $a \in A$. In this case ,the existence of vector derivative $f'(a)$ is ...
1
vote
1answer
42 views

Example of a function whose directional derivatives are always positive

I need an example of a function $f: \Bbb R^n \rightarrow \Bbb R$ such that it`s directional derivative at the direction of the vector $y$ is such that $\mathbf{D_y}(a)>0$ for a fixed vector $y$ ...
0
votes
1answer
38 views

Why does implicit differentiation apply to circle equation and works ?!

The question is, why implicit differentiation applies to equations [ Because they're not functions ] As we know, we can apply derivation to the functions, and I know that in implicit differentiation ...
0
votes
2answers
26 views

Compute a derivative using the definition and find a functional equation?

I have this function: $f(x) = f(x+y) - f(y) -x^2y - xy^2$ and i have to assume that $\lim_{x\to0}\frac{f(x)}{x}= 1$ and i already know that its derivative (using the formal definition) is: $f'(x) = ...
0
votes
1answer
44 views

derivative of ln inside ln

$y=ln(ln2x^4)$ To find the derivative of this would I have to find the derivative of the inside and then do the derivative of the entire ln function on the inside? Such as the derivative of the ...
0
votes
1answer
46 views

Limit of $\lim_{x \to 0} x^y$?

$$\lim_{x \to 0} x^y = ?$$ Can L'hopitals rule be used for for two variables? At least according to the power rule $$ \lim_{x \to 0} x^y \implies yx^{y-1} $$ Which amounts to $0 \cdot (0^{-1})$
0
votes
1answer
33 views

Differentiability of $f(x,y,z) = \sqrt{|xyz|}$

I need to study this function: $f(x,y,z) = \sqrt{|xyz|}$ at the point $P=(0,0,0)$. I have determined it is continuous at P. That is, $\lim_{(x,y,z)\to (0,0,0)} f(x,y,z) = f(0,0,0) = 0$ Also, that ...
2
votes
3answers
43 views

Logarithmic Differentiation - when to use?

Sorry if this is an ignorant or uninformed question, but I would like to know when I can (or should use) logarithmic differentiation. I haven't taken calculus in a while so I'm quite rusty. So, let's ...
1
vote
1answer
77 views

Prove the following function is differentiable

I have to prove if this function is differentiable. $$f(x,y)= \begin{cases} (x^2+y^2) \sin\frac 1{(x^2+y^2)} \iff (x,y) \neq (0,0) \\0 \iff (x,y)=(0,0) \end{cases}$$ I tried proving that all of its ...
0
votes
1answer
35 views

chain rule on trigonometric function

$y=(g(\sin(3x)))^4$ and $g(0)= 3$ , $g'(0)=\frac{1}{9}$. We are supposed to find the derivative at $x=0$. for $y'$ I tried and got $$y'=4(g(\sin(3x)))^3 \cdot ( g'(\sin(3x) + g(3\cos(3x))$$ What am ...