Questions tagged [derivatives]

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

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value)

Derivative of a function has a very natural geometric and physical interpretation: it corresponds to slope of the tangent line and to instantaneous velocity. In applications, it usually describes the rate of change of a physical variable.

Basic techniques used for computing the derivative of a given function are

It is useful to know the derivatives of elementary functions. This tag is intended for questions on the evaluation of derivatives.

Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

33197 questions

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2 answers

Double differential volume given 3 points

I am stuck on the following problem. Let $R = [0,1] \times [0,1]$. Find the volume of the region above $R$ and below the plane which passes through the 3 points $(0,0,1), (1,0,8), (0,1,2)$. I know $R$ is a square with area 1 cubit unit, and the…

derivatives

asked Jul 16 '20 at 22:02

mathjohnn

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1 answer

What is the error solving this problem about instantaneous rate of change?

I have the following problem: A hot air balloon rising straight up from a level field is tracked by a range finder $150$ meters from the liftoff point. At the moment that the range finder’s elevation angle is $\frac{\pi}{4}$, the angle…

derivatives

asked Jul 11 '20 at 20:27

ESCM

3,161

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3 answers

Applying chain rule in $f(x)=\sin(x)\cdot x\ln(x)$

Can we apply chain rule in function $f(x)= \sin(x)\cdot x\ln(x)$ What i try:: Chain rule $$\frac{d}{dx}\bigg(f(g(x)\bigg)=f'(g(x))\cdot g'(x)$$ So $$\frac{d}{dx}\bigg(\sin (x)\cdot x\ln(x)\bigg)=\sin(x)\cdot…

derivatives

asked Jun 26 '20 at 05:40

jacky

5,194

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1 answer

where is this function with exponential differentiable?

I need to find out where the function $$f(x,y)=\left\{ \begin{array}{cc} e^{\tfrac{1}{x^2+y^2-1}}, & x^2+y^2<1 \\ 0, & x^2+y^2\geq1 \end{array} \right.$$ is differentiable. Here is my progress For $x^2+y^2<1$, I can use the chain…

derivatives

asked Jun 25 '20 at 19:19

Marcos Paulo

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2 answers

What's the differentiation of $2x(\frac{dx}{dt})$ with respect to the variable $t$? Is it $2(\frac{dx}{dt})^2+2x(\frac{d^2x}{dt^2})$?

What's the differentiation of $2x\left(\frac{dx}{dt}\right)$ with respect to the variable $t$? Is it $$2\left(\frac{dx}{dt}\right)^2+2x\left(\frac{d^2x}{dt^2}\right)?$$

derivatives

asked Jun 25 '20 at 11:18

General Kenobi

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Question on how to proof two derivatives are equal dependent on two constants

Let $k \in \Bbb N_0, n \in \Bbb N, D \in \Bbb R^n, f:D \to \Bbb R,x \in \overset{\circ}{D} , \alpha \in \Bbb N_0^n$ and $c_{\alpha}, \overline{c}_{\alpha} \in \Bbb R$ with $$\lim_{\lambda \to 0} \frac{f(x+\lambda)- \sum_{|\alpha| \leq k} c_{\alpha}…

derivatives

asked Jun 25 '20 at 10:13

Drake Mass

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1 answer

How do I find $f(1)$ and $f'(1)$ if $2x+3y=5$ is the tangent of $f(x)$ at $x=1$?

Find $f(1)$ and $f'(1)$ if $2x+3y=5$ is the tangent of $f(x)$ at $x=1$. Is this correct: $$2x+3y=5$$ $$3y=5-2x$$ $$y=5/3-(2/3)x$$ From here I get that $f'(1) = -2/3$. Here I am not sure how to continue for $f(1)$? If I just replace it in…

derivatives

asked Jun 22 '20 at 18:42

congruencer

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2 answers

is $f(x,y)=(\sqrt[3]{x} + \sqrt[3]{y})^3$ Differentiable at $P=(0,0)$

$f(x,y)=(\sqrt[3]{x} + \sqrt[3]{y})^3$ and let $P=(0,0)$. the partial derivative at $P$ are $$\frac{\partial f}{\partial x}(x,y)=\lim_{h\to0}\frac{f(0+h, 0)-f(0,0)}{h}=\frac{(\sqrt[3]{h})^3}{h}=1$$ and $$\frac{\partial f}{\partial…

derivatives

asked Jun 20 '20 at 14:31

CforLinux

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1 answer

Order and degree of DE

I am not understanding how can I find the degree and order of this DE. $e^{y'''} - xy'' +y = 0$ Should I bring ln here? Please solve for me if you can.

derivatives

asked Jun 20 '20 at 04:20

Md Mosleh Uddin

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1 answer

Differential Equations

I do not understand the concept of differential in physics for example What have both the sides been differentiated by to get this result.I understand that $dz^2/dz$ gives $2z$, where has the extra $dz$ come from? Also, what has the side with…

derivatives

asked Jun 04 '20 at 07:23

Naruto

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1 answer

About differential equations

I do not understand the concept of differential enter image description here for example What have both the sides been differentiated by to get this result.I understand that dz^2/dz gives 2z,where has the extra dz come from also what has the side…

derivatives

asked Jun 04 '20 at 08:21

Naruto

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1 answer

Partial derivatives need help?

I have to find the derivative $dw$ if $w=F(u,v,z)$ where $u=x^2+y^2, v=x^2-y^2$ and $z=2xy$. So $dw=\frac{\partial w}{\partial z}dz + \frac{\partial w}{\partial u}du + \frac{\partial w}{\partial v}dv$. How to find $dw$ now?

derivatives

asked Apr 23 '13 at 09:47

fdsf

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1 answer

How to calculate a directional derivative

If I have a function like f(x,y,z) = xyz + 1, it´s easy to calculate the directional derivative by ∇f•v. But if we have a vector f(x,y,z)= (2xy, xz², 3y²z), ¿how we can calculate his directional derivative in the direction v(1,4,4) in the point…

derivatives

asked May 28 '20 at 00:01

Laxus D

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1 answer

Maximum area without use of derivative

In a recent question here, the area of an inscribed yellow rectangle is shown maximum when OB bisects the central sector angle POQ using differentiation. Is it possible to arrive at this result without finding derivative by logic of increasing and…

derivatives

asked May 27 '20 at 04:08

Narasimham

40,495

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2 answers

What are differentials with respect to?

If $x=y^2$, then $\frac{dx}{dy}=2y$, which means that the rate of change of $x$ with respect to $y$ equals $2y$. We can also find the differential of $x$. We get that $dx=2ydy$. But what is this with respect to? It doesn't make any sense to have a…

derivatives

asked May 13 '20 at 18:38

James Ronald

2,331

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