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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derivative of double integral w.r.t. y(x)

I'm reading Bishop "Pattern Recognition and Machine Learning" and cannot understand how to get the following result: Imagine we have $\mathbb{E}[L] = \int\int (y(x) - t)^2p(x, t)dxdt$ and want to minimize it choosing y(x). So, we want…

derivatives

asked Dec 28 '22 at 12:08

asukaev

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1 answer

Basic question about differentiation

The question is: Let $F(x) = f^2(g(x))$. If $g(1)= 2, g'(1)= 3, f(2) = 4$, and $f'(2) = 5$, find $F'(1)$. $$ F'(x) = f'(f(g(x)) * (f \circ g)'(x) = f'(f(g(x)) * f'(g(x)) * g'(x)$$ put $1$ into $F'(x)$: $$F'(1) = f'(f(g(1)) * f'(g(1)) *…

derivatives

asked Dec 10 '22 at 11:36

davislui

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Slope of a points in a Circumference

I need to compute the slope (m) of the nine points of a circumference divided in equal parts. But a circumference is not a function. I make a Geometric approach but I am not satisfied with it. Do anyone know how to solve it analytically.

derivatives

asked Aug 03 '13 at 13:54

Haizum Skallah

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Derivatives of $n_{0}\left( x\right) =e^{y\left( x\right) }$ in terms of the derivatives of $y\left( x\right)$

Given function $n_{0}\left( x\right)=e^{y\left( x\right)}$ where $y\left(x\right)$ is an arbitrary function, what are the multiple derivatives of $n_{0}\left( x\right)$ in terms of the multiple derivatives of $y\left(x\right)$? For example:…

derivatives

asked Nov 22 '22 at 02:18

Edward L. Ruden

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1 answer

Derivative of $f(x) = a \sum_{i=1}^x 10^{x-i}$

Consider the following function: $f(x) = a \sum_{i=1}^x 10^{x-i}= a (10^{x-1} + 10^{x-2} + \cdots + 10^{x-x})$ whose domain are the positive integers greater than 1. What is its derivative function? Despite the function being non-continuous I think…

derivatives

asked Aug 01 '13 at 14:57

Ernest A

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Why are the derivatives of $\frac{x^2+1}{x^2+x+1}$ and $\frac{-x}{x^2+x+1}$ the same?

Why are the derivatives of these functions the same? $$\frac{x^2+1}{x^2+x+1} \qquad\qquad \frac{-x}{x^2+x+1}$$ original exercise text (See part (e).) I have tried to answer this question and consulted the answer booklet but this did not make much…

derivatives

asked Oct 22 '22 at 12:49

strugglingmathematician

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What are the critical points of $f$ given its derivative is $f'$?

The derivative of the function $f$ is, $$ f'(x)=\frac{x^2(x-1)}{x+2},~ x\ne-2. $$ The obvious critical points are $x=0$ and $x=1$. However, since $f'(-2)$ is not defined, can $-2$ be a critical point? I have read that for $c$ to be a critical…

derivatives

asked Oct 03 '22 at 15:17

rayank97

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least squares derivatives

Here is my problem: In problem (a), I thought that $\frac{\partial}{\partial\theta_i}l_i(\theta)=(\sum_{k=1}^{p}x_{ik}\theta_k-Y_i)x_{ij}$ Thus $\nabla_{\theta}l_i(\theta)=[\frac{\partial}{\partial\theta_1}l_i(\theta) \;…

derivatives

asked Sep 01 '22 at 16:06

SeungJae Bang

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1 answer

product rule for multiple products

Consider the following derivative $$ \frac{d}{dx}\left[\prod_i\left(x\prod_jf_{ij}(x)\right)\right] $$ I am not certain how to perform this. Can I apply the product rule as follows: Let $g_i=x\prod_j f_{ij}(x)$…

derivatives

asked Jul 15 '22 at 13:12

RedPen

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If derivative of $e^{ax} \cos{bx}$ with respect to $x$ is $re^{ax}\cos(bx + \tan^{-1} \frac {b} {a})$

Problem: If derivative of $e^{ax} \cos{bx}$ with respect to $x$ is $re^{ax}\cos(bx + \tan^{-1} \frac {b} {a})$. Then find $r$ when $a>0,b>0$ Solution: Differentiating $e^{ax} \cos{bx}$ w.r.t $x$ we get $ ae^{ax} \cos{bx} -be^{ax} \sin{bx} $ But I…

derivatives

asked Jul 22 '13 at 09:38

rst

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Vector by vector derivative

I want to solve the derivative of $\mathbf{f}=\mathbf{a}^T\mathbf{x}\mathbf{c}^T$ with respect to $\mathbf{x}$, where $\mathbf{a}$ and $\mathbf{x}$ are $M\times 1$ vectors, and $\mathbf{c}$ is a $N \times 1$ vector. $\mathbf{a}$ and $\mathbf{c}$ are…

derivatives

asked Jul 08 '22 at 05:33

ZYX

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What does "derive with respect to $\arccos\left(x^2\right)$" mean?

Derive with respect to $\arccos\left(x^2\right)$ where $$f(x)=\arctan\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right)$$ Can someone please explain to me what does "with respect to $\arccos\left(x^2\right)$" mean ?

derivatives

asked Jun 18 '22 at 20:22

Mo2edmatic

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Is the derivative of a function the secant line?

I am just learning derivatives and I found the derivative of $4x-x^2$ to be $4-2x$. At point $(1,3)$ the tangent line is $2x+1$. Now when I graph this, the derivative $4-2x$ cuts through the function $4x-x^2$. Does that mean the derivative is the…

derivatives

asked Jul 17 '13 at 03:11

user5826

11,982

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1 answer

Find m so that f has one local extreme point.

Let $f\colon (-1,1) \to \mathbb R $ and $ f(x)=e^x(x^2+x+m) $. The function f has only one extreme local point if and only if m belongs to the set: a) (-5, 1); b) {-5, 1}; c) [-5, 1); d) ${\frac{5}{4}}$. I tried to calcule the derivate…

derivatives

asked May 04 '22 at 23:16

claudiubagiu

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1 answer

Rate of change of surface area of a sphere given the rate of change of the radius

Air is pumped into a spherical balloon such that the radius of the balloon increases at the rate of $\dfrac{1}{20}\pi$ cm/s when the radius is $8.5 \text{cm}$. Find the rate of change of the surface area of the balloon at this instant. How do I do…

derivatives

asked Mar 21 '22 at 05:36

user1039203

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