Mathematics Stack Exchange
Mathematics Stack Exchange

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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Derivative of a function with condition

Consider a function $$f(x,y)= \begin{cases} xy, & |x|<|y| \\ 0, & \mathrm{else} \end{cases} $$ Let's say, we are going to compute $f_{,x}(0,0), f_{,y}(0,0), f_{,xy}(0,0)$ and $f_{,yx}(0,0)$ in point $(0,0)$. Important note: I took the liberty of…
user74200
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About rate, my ideia is wrong?

The following problem: An oil tank should be drained for cleaning. V oil gallons are left in the tank T minutes after the drain started, where $V = 40*(50 - t^2)$ So, I change to: $f(t) = 40*(50 - t^2)$ a) the median rate which the oil is drained…
mastergoo
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Checking differentiability

I have a question to share. Find the points at which the following function is not differentiable: $$f(x)=\max \lbrace 1-x,1+x,2\rbrace \quad\forall x\in \mathbb{R}$$ I have joined this site recently and do not have enough reputation. So I am…
Debashish
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How do you determine such derivatives can be simplified?

Background When working out the Laplacian in spherical coordinate system, either via chain rule or differentiating the basis vectors, one arrives at the following: $\nabla^2 = \frac{\partial^2}{\partial r^2} + \frac{2}{r}\frac{\partial}{\partial r}…
syockit
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Second Derivative of log

Let: $\log(s)=z$ I understand that $$\frac{\partial}{\partial s}=\frac{\partial}{\partial z} \frac{\partial z}{\partial s} = e^{-z}\frac{\partial}{\partial z}$$ What is the second derivative, ie $\frac{\partial^2}{\partial s^2}$ Applying the…
Anastasiosyal
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derivative of $y=(x^2+x^3)^4$

I can't figure out where I am going wrong. $$y=(x^2+x^3)^4$$ chain rule it first $$4(x^2+x^3)^3* \frac d{dx}(x^2+x^3)$$ which should become: $$4(x^2+x^3)^3(2x+3x^2)$$ factoring out should give me: $$4*x^2*x(1+x)^3(2+3x)$$ which to me says the answer…
Joshhw
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How to show where a funcion is differetiable?

I mean what is the basic technique on functions which are defined like $h(x)$= {$f(x)$ if $x$$\in$$[a,b]$ and $g(x)$ if $x$$\in$$[c,d]$}. An example:$f(x)$={$x$*$cos({1\over x})$ if $x$ is not $0$ and $0$ if $x=0$}.
user128576
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Stationary point question

Find the coordinates of stationary points on the curve with the equation $(y-2)^2e^x=4x$ I differentiated to get $2(y-2)e^x\frac{dy}{dx}+(y-2)^2e^x=4$ $\frac{dy}{dx}=0$ $(y^2-4y+4)e^x=4$ What do I do from here?
Jim
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Differentiation calculation

If $a, b, p$ and $q$ are positive with $a
Jim
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Derivative of $\ z=v^{3}u^{5} $ by chain rule and substitution

Let $\ z=u^{3}v^{5} $ where $\ u=x+y, v=x-y $ Find $\ \frac{dz}{dy} $ For that I just did $$\ \frac{dz}{dy}=\frac{dz}{du}\frac{du}{dy}+\frac{dz}{dv}\frac{dv}{dy} $$ And I got: $$\ 3(x+y)^{2}(y-5)^{5}+5(x+y)^{3}(x-y)^{4}$$ Is this right? This is the…
Javittoxs
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What justifies using the derivative $\frac{\mathrm{d}y}{\mathrm{d}x}$ as a fraction?

For instance, one method of solving first-order equations uses separation of variables. $$\frac{\mathrm{d}y}{\mathrm{d}x}=yx$$ $$\frac{\mathrm{d}y}y=x\;\mathrm{d}x$$ $$\log |y|=x^2/2+C$$ $$y=Ce^{\frac{x^2}2}$$
user87611
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Series differentiation

$\displaystyle e^x= \sum_{j=0}^{\infty} \frac{x^j}{j!}$ The textbook says that when we differentiate this, we obtain the same series, so that $(e^x)'=e^x$. But why is this? Isn't the derivative $\displaystyle \sum_{j=0}^{\infty}\frac{jx^{j-1}}{j!}$?
kiwifruit
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Differentiability of $\sum x^j$

Prove that $\sum x^j$ is differentiable on (-1,1), and $$\frac{d}{dx} \sum x^j = \sum (j+1) x^j$$ I am able to prove that $\sum x^j$ converges uniformly to $\frac{1}{1+x}$. However, how do I get this derivative? It doesn't seem to follow traditional…
kiwifruit
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applications of derivatives : maxima and minima

To finding the the maxima and minima why do we equate the derivative of a function with zero and n0t with any other number like 10,100 ?
akanksha
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A basic question on successive differentiation

How to prove that $$\frac{d^r}{dx^r}\cos x + i\frac{d^r}{dx^r}\sin x = i^r e^{ix}\ ?$$ I can understand it by putting values, but how to prove it?
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