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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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Differential identity $\left(x^2\frac{d}{dx}\right)^nf(x)=x^{n+1}\frac{d^n}{dx^n}\left(x^{n-1}f(x)\right)$

I have found the following differential identity: $$\left(-x^2\frac{d}{dx}\right)^nf(x)=(-1)^n x^{n+1}\frac{d^n}{dx^n}\left(x^{n-1}f(x)\right)$$ I have used it to find an alternative Rodrigues representation for Bessel polynomials. My proof we have…
giorgiomugnaini
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Proof that derivative operator cannot be written in terms of composition operator (without limits)

Difference delta operator can be written without a limit: $$\Delta[f(x)]=f(x+1)-f(x)$$ The same is true for any other finite difference operator. But what about derivative? Is there a proof that it cannot be expressed similarly without a limit? Note…
Anixx
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Prove $f'(0)=0$, if $|f(x)|≤x^2$

Let $f:(-a,a) \longrightarrow R$, $a>0$. Such that $$ |f(x)|≤x^2 $$ What I did was taking out the module bars so I get $-x^2≤f(x)≤x^2$ and I see that at $x=0$ the function must be zero. I see why f'(0) must equal zero at that point, but I have no…
YoTengoUnLCD
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Continuity of derivative implies differentiability at a point

Suppose $I\subset\Bbb{R}$ is an open interval, $a\in I$ and $f:I\to\Bbb{R}$ is continuous. Suppose also that $f$ is diffble on $I-\{a\}$. Show that if $\lim_{x\to a} f'(x)=s$ exists, $f$ is diffble at $a$ and $f'(a)=s$. I honestly don't know where…
sun
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Unbounded derivatives on a bounded set

Suppose that $f$ is differentiable on a finite interval $(a,b)$ and $$ \lim_{x\to a^+}f(x)=\lim_{x\to b^-}f(x)=\infty. $$ Prove that, for any $r\in\mathbb{R}$ there exists $c\in (a,b)$ such that $f'(c)=r$. MY THOUGHTS: This theorem seems painfully…
Laars Helenius
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Differentiability implies continuous derivative?

We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be defined as differentiable. However in the case of 1…
Kelvin S
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Prove that a function diverges to infinity if its derivative has a positive lower bound for all $x$ on a closed ray $\left[ a,\infty \right)$.

Let $f$ be differentiable on $\left[ a,\infty \right)$. Prove that if $\exists m>0\,\forall x\in \left[ a,\infty \right)\,f'\left( x \right)\ge m~$, then $\lim\limits_{x\to\infty}\,f\left( x \right)=\infty $. I began by using the average value…
Gad Maor
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Please explain the algebra in the last part of derivative of the sigmoid function

http://www.ai.mit.edu/courses/6.892/lecture8-html/sld015.htm how does this: $${1\over 1 + e^{-x}} \cdot {-e^{-x}\over 1 + e^{-x}}$$ become this: $${1\over 1 + e^{-x}} \cdot \left (1 - {1\over 1 + e^{-x}}\right)$$
user3234550
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12th order derivative using Leibniz

I don't fully understand the rule of Leibniz and I'm trying to find the 12th order derivative of: $x\cos \left(x\right)$ How do I find this using the Leibniz rule?
Stanko
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Prove the following equality regarding partial derivatives

Let $f:\Omega\subset\mathbb{R^2\to\mathbb{R}}$ be a function such that $f\in\mathit{C^1}(\Omega)$. Now, consider the function: $$g(x,y,z):=x^4f(y/x,z/x)$$ Prove that $$x\frac{\partial g}{\partial x}+y\frac{\partial g}{\partial y}+z\frac{\partial…
F.Webber
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Find the $n^{th}$ order derivative of $x^n \ln x$

I'm doing it completely wrong, I'm sure, but I'll still show my attempt: $n^\text{th}$ order derivative of $x^n$ is $n!$ and of $\ln x$ is $(-1)^{(n-1)} (n-1)! x^{-n}$ So, using Leibnitz rule I got $n^\text{th}$ order derivative of $x^n \ln x$…
Diya
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Does all function's domain stay the same\expands as we derivate them?

Lets define a funciton $f(x)$ with a domain of, lets say $a>x>b$. If I derivate this function, it's domain will always stay the same or expand? Or it can be "reduced"? Is that mean that $f'(x)$ must be defined in the following domain?: $$g>x>t$$ $$g…
Eminem
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How to find 50th derivative of $\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$?

I need to compute 50th derivative of $$\left(\dfrac{\sqrt{1-x}}{\sqrt{1+x}}\right)$$ Of course I would not compute 50 derivatives. I want to find a certain regularity. And what I have: As can be seen, there is a certain regularity. But I can not…
martin
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differentiate *g(x)* if $g(x)=e^xf(e^{-x})$

differentiate g(x) if $g(x)=e^xf(e^{-x})$ Using any website to evaluate this derivative like wolframalpha.com we will get the result ===> $e^xf(e^{-x})-f'(e^{-x})$ But we know from the Product Rule of derivatives that it will like…
Maher
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How the derivative might fail to exist

Can a function have both a vertical tangent and cusp? Does The function $3x^{1/3}(x+2)$ have a vertical tangent and if so why? I believe that it has a cusp.
user159778
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