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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.

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The n-th derivative of $x \cdot \cos(2x)$

I have a function $f(x) = x \cdot \cos(2x)$. I have to find the $n$-th derivative $f^{(n)}(x)$. I know I have to use Leibnitz's formula, but how to get a general formula for this $f(x)$?
Andrej
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Can't undestand $\frac{d}{dx} \left(c + x^2 + \frac{1}{2} \cos(2x) \right)$

I'm trying to solve $\frac{d}{dx} \left(c + x^2 + \frac{1}{2} \cos(2x) \right)$ but I'm stuck at the part that uses the chain rule to find the derivative of $\cos(2x)$. My solution considers that $\frac{d}{dx} (\cos(2x))$ is $-2 \cdot \cos(2x)$, but…
juliano.net
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Can differentiation be defined in an algebraic way?

Is it possible to define the operation D of differentiation of real functions in an abstract way, as for example by the fundamental properties of the derivative: D(f+g) = D(f) + D(g) D(fg) = fD(g) + gD(f) if f(x) = x, D(f) = 1 (to avoid the…
exp8j
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Use implicit differentiation to determine increase/decrease of function?

The derivative of the equation $x^2-y^3=4y+9$ can be found using implicit differentiation to yield $y'=2x/(4+3y^2)$. (I think.) I am tasked with showing where the original function is rising, falling, and resting using the equation for its…
Omne Bonum
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Nth derivative differentiation

The question is the nth derivative of $x^{x}$ I have try solving this nth term differentiation of x raised to the power of x. Using $y=x^{x}$ and taking he log of both sides...but I don't know how to differentiate the nth term. Pls help
Olaniyi Micheal mc olas
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$y' = a(y) \cdot y$ yith $a$ defined in an inequation

I was wondering if there was knowledge to the following differential equation: $$y' = a(y) \cdot y$$ with $a$ is defined as being $a_0$ when $y < b$ and $a_1$ elsewhere with $a_0 , a_1, b$ reals. The family of functions is obviously…
PackSciences
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How to show $f(x,y)=|xy|^\alpha\log(x^2+y^2)$ is differentiable at $(0,0)$?

Show that if $\alpha > 1/2$, then $$f(x, y)=\begin{cases} |xy|^\alpha\log(x^2+y^2), ~(x, y) \ne (0,0)\\\\ ~~~0, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~\text{otherwise}\;. \end{cases}$$ is differentiable at $(0,0)$.
JFK
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How does concavity look like for non-differentiable or discontinuous function?

In my university textbook, it shows that if a function is concave downwards then it's graph looks something like this https://en.m.wikipedia.org/wiki/File:ConcaveDef.png But what if a function is discontinuous or continuous but non-differentiable.…
William
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Gradient of marginal likelihood of Gaussian Process w.r.t likelihood parameters with Laplace approximation

The derivation of gradient of the marginal likelihood w.r.t covariance function hyperparameters $\theta$ is given in http://www.gaussianprocess.org/gpml/chapters/RW5.pdf, page 125. However, the gradient w.r.t likelihood parameters (let's call them…
Truong
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Basic differentiation problem using multiple rules

$$\text{Find}~ \frac{\,dy}{\,dx}~, \qquad \text{where}~~y =\frac{\sqrt{2x^2}}{\cos x}$$ Here's the basic question. The solutions suggest to use the quotient rule for the top half and bottom half and to use the chain rule on the top half of the…
JKong
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Calculate $\lim_{x\to \infty}(x + \ln(\frac{\pi}2 - \arctan(x))$ using L'hopital's rule

I'm new to L'hopital's rule. I know i need to convert it to $\frac{\infty}\infty$ or $\frac{0}0$. But I have no idea how to convert the following equation. Thanks in advance for your help! $$\lim_{x\to \infty}(x + \ln(\frac{\pi}2 - \arctan(x))$$
Hellowhatsup
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Computing the gradient of a function

Function: $f(x,y,z) = e^{-x} (x^2 + y^2 + z^2)$ I need to differentiate this equation for $f_x, f_y$ and $f_z$: $\nabla f = \langle f_x, f_y, f_z \rangle$ What I Got: \begin{align} f_x &= −e^{−x}(x^2+y^2+z^2)+e^{−x}(x^2+y^2+z^2)2x \\ f_y &=…
user430574
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(partial) Derivative of norm of vector with respect to norm of vector

I'm doing a weird derivative as part of a physics class that deals with quantum mechanics, and as part of that I got this derivative: $$\frac{\partial}{\partial r_1} r_{12}$$ where $r_1 = |\vec r_1|$ and $r_{12} = |\vec r_1 - \vec r_2|$. Is there…
Filip S.
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When is the second derivative the same as the product of two first derivatives?

In the book Mathematical Methods for Physics and Engineering by RIley Hobson and Bence, an example is given in chapter 5 of using partial differentiation. The example problem Now the first partial derivatives are evaluated, which appears perfectly…
ekke
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How does one take the derivative of the Electric Field equation?

How do you take the derivative of this? k and r are constants $E=kq\hat{r}(\frac{1}{r^{2}})$ $\frac{d}{dq}(E)=\frac{d}{dq}(kq\hat{r}(\frac{1}{r^{2}}))$ $\frac{d}{dq}(E)=(\frac{k\hat{r}}{r^{2}})\frac{d}{dq}(q)$ This is where I am now. And the answer…
Hack Delta
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