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 variables in terms of other variables

Sorry for the confusing title, but I'm doing economics homework and I got snagged on something strange in my brain. Suppose I have a variable $y = x^2$. I have the expression $$p = \frac{C'(x) y}{y-1}$$ If I want to find $\frac{d p}{d y}$, do I have…
Alex Peniz
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The Differentiability of $f$ at $x=0$

Suppose $f(0)=0$ and $\lvert f(x)\rvert> \sqrt{\lvert x\rvert}$ for all x. Show that $f^\prime(0)$ does not exist. (Adam's Calculus). Instead of going fot the definition of differentiation, I tried to show that the limit does not exist at $x=0$. By…
Nicki Bood
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derivative $f(z,y) = g(x+y)h(y)$ with respect to $y$

Suppose I have differentiable function $g(x+y)$ and $h(y)$ and define $z = x+y$. Also, I assume that for any $x_1,y_1$ there exists $x_2,y_2$ such that $g(x_1+y_1) = g(x_2+y_2)$. Then, I want to take derivate of $$f(z,y) = g(x+y)h(y)$$ with respect…
user1292919
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derivative dot(newton) notation $\frac{d}{dt} \dot{x}^2$

I can't find how to derive $\dot{x}^2$ I know $\frac{d}{dt} \dot{x} = \ddot{x}$ My guess $\frac{d}{dt} \dot{x}^2 = 2\ddot{x}$, but I'm not sure.
proxima
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Question of a function with two unknown variables.

So the question goes like this: Equation $6xy^{3}+4y-5x^{2}=6$ will define a plot. When we limit the domain and codomain we get a function $y=f(x)$. Find a common expression for the derivative $y'$. I can not figure out the $y=f(x)$ part of this…
f1tz
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Question about the second derivative test of a function of two variables.

i understand that the second derivative test of a function of two variables uses the value of a function that utilizes $f_{xx}f_{yy}-f_{xy}^2$. I would just like to ask what conclusion would i be able to derive if $f_{xx}$ is undefined at a certain…
Ross Henderson
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$Q=Q_1+Q_2,Y=a\times {Q_1}^m+b\times {Q_2}^n$: how to take derivative w.r.t. the sum of two variables

How to take the derivative with respect to the sum of two variables? $Q=Q_1+Q_2$ $Y=a\times {Q_1}^m+b\times {Q_2}^n$ Can we take the derivative of $Y$ with respect to $Q$? Can this be solved using this method?
cclinoom
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Differentiability, parameter a

I have a function: $$ f(x) = \frac{\arctan(x)^2}{\ln(1+x^2)} \text{when x $\neq$ 0} \\ a ~~\text{when x =0} $$ Using Taylor series, I have calculated that the function is contiunous at $ a= \frac{1}{3} $. However, I am not sure, how to prove its…
Funny
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Given $F(y)$, what is $F(y+dy)'$?

How to find $F_y(y+dy)'$? Particularly I'm trying to learn how to work with Random Variables. But this question is on pure differentials. I know there is $f(g(x))'=f'(g)g'(x)$, but what is $(y+dy)'=$? I guess $(y+dy)'=0+something$. But what is…
Vitali Pom
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Not able to understand the simplification done here.

How do we get from $(1)$ to $(2)$ ? $$\frac{\partial}{\partial x} \frac{f(x,y)}{G(y)}+\frac{\partial}{\partial y} \frac{f(x,y)}{G(y)}=0 \tag{1}$$ $$ \frac{f(x,y+z)}{G(y+z)}=\frac{h(x-z,y)}{G(y)}, \text{for } x \geq z\tag{2}$$
Cookie4free
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Derivative $\frac{dt}{dx}$ using $\frac{dx}{dt}$

Suppose I have a function $x(t)$ such that $\frac{d}{dt}x=x(t)y(t)$. Can I affirm hat $$\frac{dt}{dx}=\frac{1}{\frac{dx}{dt}}=\frac{1}{x(t)y(t)}$$ I am sorry but I am a bit confused about this argument
user268193
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The landing velocity of an airplane is 100 mi/hr. Constant deceleration and comes to a stop after traveling 1/4 mile. Find the deceleration.

Please could someone confirm that my calculations and answer is correct. The landing velocity of an airplane (i.e., the velocity at which it touches the ground) is 100 mi/hr. It decelerates at a constant rate and comes to a stop after traveling …
OpenSauce
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Determine for which x the derivative exists of: $f(x)=\ln|\sin(x)|$

I have the following function: $f(x)=\ln|\sin(x)|$ I've caculated the derivative to: $f'(x)=\frac{\cos(x)}{\sin(x)}$ And the domain of $f(x)$ to: $(2\pi n, \pi+2\pi n ) \cup (-\pi + 2\pi n, 2\pi n)$ And the domain of $f'(x)$ to: $(\pi n, \pi+\pi n…
freya
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Derivative of variables vs functions

Edit: Removed specific equation. General question is: How are variables and functions treated differently during differentiation?
BearsBeetsBattlestarGalactica
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Differentiating $u(b, t) = u(b + ch, t + h)$

I'm a beginner in PDE, studying the introduction part of Strauss' Partial Differential Equations book. I'm stuck in a trivial part that says: $u(b, t) = u(b + ch, t + h)$ Differentiating this with respect to $h$ and putting $h = 0$, we get $0…
stoneaa
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