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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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Taking the Derivative of the Gaussian Normalizing Condition with respect to $\sigma^2$

In Bishops book, "Statistical Pattern Recognition", there is one exercise, which states to derive the second order moment of the Gaussian Distribution: $E[x^2] = \int_{-\infty}^{\infty} N(x|\mu, \sigma^2 )x^2dx = \mu^2 + \sigma^2 $ As a hint, the…
kklaw
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How to prove that $T \frac{d^2 T}{d x^2} = T f \frac{df}{dT}$

Considering that: \begin{equation} f(T) = \frac{dT}{dx} \end{equation} and: \begin{equation} T \frac{d^2 T}{d x^2} = 0 \end{equation} \begin{equation} T f \frac{df}{dT} = 0 \end{equation} how to prove that \begin{equation} T \frac{d^2 T}{d…
efirvida
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How do I apply the differential to points in $\Bbb{R}^2$?

I know how to compute the differential, for example, for $f(x,y)=(x^2,y^2)$, we obtain: $$f'=\left( \begin{array}{cc} 2 x & 0 \\ 0 & 2 y \\ \end{array} \right)$$ Now I want to apply it to points in $\Bbb{R}^2$. What do I do? First choose values…
Red Banana
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Differentiation of $x$ to the power of $y$ with respect to $x$

As the title suggests, I need to differentiate $x$ to the power of $y$ with respect to $x$. Not sure how to start. Do I need to take natural log on both sides? That is: $\dfrac{d}{dx}x^y=?$
lakshmen
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How is it that the second derivative of a function can be 0 at a maximum?

How is it that a maximum of a function can have the second derivative as 0? I thought the gradient of the tangent to the curve changed from positive to negative at a maximum and is therefore decreasing.
overthetop
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Prove of nonzero partial derivatives

I am dealing with the following problem of multivariate calculus but I haven't been able to solve it yet: "If $u$ and $v$ are class 2 functions in $\mathbb{R}$ and $z=z(x,y)$ is class 2 in $\mathbb{R^2}$,verifying that…
CharlesJA
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Does every point of a differentiable have a unique tangent i.e does every point of a non-linear differentiabel function have different derivatives?

I want to ask whether every point of a differentiable nonlinear function can have unique tangents with respect to its neighbouring points i.e every point of a differentiable function have different derivatives than its neighbouring points…
Anuj
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find the derivative of $f(x)=\frac1x+2$ using definition of the derivative

I am struggling to find the derivative of $f(x)=\frac1x+2$ (and/or $f(x)=\frac1{x+2}$) using the definition of the derivative. thank you
pashy
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What happens inside when differentiating composite functions

Let us take a function $f(x)$ and another function $g(f(x))$. Suppose we are interested in finding $\frac{d(g(f(x)))}{d(f(x))}$. What we high schoolers are taught is suppose $f(x)=u$ and then find $\frac{d(g(u))}{du}$ that is we treat $f(x)$ as a…
madness
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How do I state the x-coordinates of the inflection points of the curve below.

I was doing my HW and I encountered a problem that confused me greatly. I will try to show an image because the question gives a graph and asks us to find the inflection points for the curve on the f , the f', and the f''. So yeah. I am very…
Mehmed Sahan
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Directional derivative of $x \mapsto A(x):= x_1 A_1 + x_2 A_2 + \dots x_n A_n $

Let $\mathcal{S}_{m \times m}$ denote the space of real valued symmetric $m \times m$ matrices. Suppose $A_1, A_2 , \dots, A_n \in \mathcal{S}_{m \times m}$ are such symmetric $m \times m$-matrices.Now consider $A: \mathbb{R}^{n} \to \mathcal{S}_{m…
mathsstudentTUD
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monotonicity of the remainder term of exponential

Let $\exp(x)=\sum_{n=1}^\infty \frac {x^k}{k!}$ be the exponential function defined on real numbers. It is trivial that $\exp(x)$ is monotone increasing. However, let $h_1(x)= \frac {e^x-1}{x}$, $h_2(x)=\frac {e^x - 1 - x}{x^2}$, so on and so forth.…
Henry Davii
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Must all differentiable $f$ approximately behave linearly like $f'(p)$ in a small neighborhood of p?

I saw that the function $f(x) = {x \over 2} + x^2 \,\text{sin}({1 \over x})$, with $f(0)$ defined as $0$, was used as an example to show that even though a function is differentiable and the derivative is positive at the origin, the function is not…
teepung
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Is there any interpretation for the ratio of the second derivative to the first derivative?

Suppose we have a differentiable function $g(x)$. Let's define a function $h(x)=axg(x)$ where $a$ is a constant (this is just as an example). I want to check the convexity/concavity of function $h(x)$. Is there any interpretation for the ration of…
Amin
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Show that a derivative is positive

Given $r_1, r_2 \in \Bbb R^+$, let the function $f : \Bbb R^+ \to \Bbb R^+$ be defined by $$f(x) = \frac{1- \text{exp}(-(r_1 +r_2)x)}{1-\text{exp}(-r_1x)}$$ I need to show that $f$ is increasing in $x$ for all $x > 0$. I've tried obviously to…
Matteo Bulgarelli
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