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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Partial derivatives math problem?

I have the function $u=\ln(1+x+y^2-z^2)$. I have to find the value of the expression $u'(x) +u'(y) +u'(z)$ for $z=y=x=1$. In a usual case, I keep one of the variables constant, how do I keep two here?
Tweetingbird
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derivative when f(x+dx) is hugely different from f(x)

I looked at the geometric meaning of derivatives and the chain rule video in the 3blue1brown calculus series. https://youtu.be/YG15m2VwSjA My question is that when we calculate the derivative of the product of two functions, we disregard the little…
Khanna111
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Partial derivative. Equation of motion

Would somebody,please, explain to me the following notation: $$\sigma_{xx}+d\sigma_{xx}=\sigma_{xx}+(\partial\sigma_{xx}/\partial x)\,dx. $$ I do not understand how $d\sigma_{xx}$ transforms to partial $(\partial\sigma_{xx}/\partial x)\,dx.$ Thank…
user504068
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Differentiating inequality

So I was solving a question that goes as follows: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and satisfy $$ \left| f(x) \right| \leq x^4 + 5x^2\qquad \forall x $$ Show that $f'(0) = 0$ T.P.T means to prove that Now I know we can't…
B Luthra
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How to relate perimeter to area using differentiation?

For reference I started learning differentiation only a few days ago. Question; https://i.stack.imgur.com/pdBRW.jpg I understand how things work when they are in numbers e.g polynomials, but am completely lost as to how I can apply differentiation…
janes
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What does it tells us if first derivative is a parabola?

I'm dealing with a specific polynomial function. The first derivative of it is displayed below. As you can see, it has the shape of an asymetric function. But what does this tells us about the initial function in general and in terms of finding a…
Steven01123581321
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How to calculate a linearly decreasing beep interval for $X$ beeps in $t$ seconds?

I want a buzzer to beep $X$ times in $t$ seconds, but I don't want it to have a regular interval (something like $t/X$). Instead, I want the interval to linearly decrease, so it "beeps faster" as it approaches the end. Each beep lasts $b$ seconds. I…
Micael Jarniac
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Can we multiply any rate of change with dt/dt?

Suppose that we have dy/dt in an equation like: dy/dx=x Can we write this equation as dy/dx*dt/dt=x? Moreover, can we always multiply any derivative with dt/dt?
Shanif Ansari
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How to solve $y’’(t)=y(t)+t$

How to solve the ODE $$ y’’(t)=y(t)+t, $$ with initial conditions $y(0)=0$ and $y’(0)=0$. I’ve found $y=-t$ and $y=e^t-t$, but both aren’t valid solutions because of the two initial conditions.
Maurits van Altvorst
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Derivative of $x^{\sin x}$

Why $ \sin x.x^{\sin x-1}\cos x$ wrong? Why I cannot treat $\sin x$ just like usual power $x^a = ax^{a-1}$
Dini
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Sketching the derivative a graph with asymptotes.

I have to sketch the derivative of a particular graph (see attached image), however, it seems my answer is wrong, but I do not understand why. If anyone could help explain why I’m wrong, that would be great!
Jamminermit
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Total differential

I am considering a function $f(x,y)$ with all the appropriate assumptions so that what comes next is well defined. I think we have the equivalence: $$\mathrm{d}f=\frac{\partial f}{\partial x}\mathrm{d}x+\frac{\partial f}{\partial…
pluton
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Principle of least action (differentiability)

If I fix $x \in M$ and $y \in M_0$, how do I show that the mapping $g: \mathbb{R} \rightarrow \mathbb{R}$ given by $ g(s) := I[x+sy], \hspace{20pt} s \in \mathbb{R}$ is differentiable and what is then the derivative $g'(0)$ ? I have the…
StudentP
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I want to find the rate of change of height of water in this question

An empty container of length $12 \ \textrm{cm}$ and the vertical cross section of it is an isosceles right angled triangle. If the water poured in it at a rate $\frac{10 \ \textrm{cm}}{\textrm{sec}}$ find the rate of change of height of water …
Mishoo Maher
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Derivative and difference (intervals) between values

Having only a high-school math background (~30 years ago), I'm having issues with the idea of derivative. I understand that it's to be considered as a sort of "acceleration" of the values of a function, i.e. the rate at which such values change.…
ChatterOne
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