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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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Approximation of derivative using combinations

I have sets $A = \left\{a_{1},a_{2},a_{3},\dots,a_{n} \right\}$ and $B = \left\{b_{1},b_{2},b_{3},\dots,b_{n} \right\}$. If they are time-series sorted by their indices, I can take the differences, $$ d = \frac{\Delta A}{\Delta B} =…
user2167741
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Computing derivative $x^{x^x}$

Could you show me how to compute $f'(x)$, where $f(x)=x^{x^x}$. I know that for $g(x)=x^x=e^{x\ln x} \ \ $ $g'(x)=e^{x\ln x}(\ln x+1)$ Now, my problem is this: is $f(x)=x^{x^x}= e^{x^x \ln x}$ or $e^{x\ln x^x}$ ?
Hagrid
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Find $\lambda\in\mathbb{R}$ such that $y=e^x$ and $y=\lambda x^2$ touch.

Find $\lambda\in\mathbb{R}$ such that $y=e^x$ and $y=\lambda x^2$ touch. I'm just a beginner on derivatives, and I guess it should be done using them, but I'm totally stuck. (I guess it's not "touch" in English but oh well.)
Lazar Ljubenović
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Doubt about temporal derivative of a partial derivative

I have $\theta (t)$ and $\phi (t)$ and I have to find: $$\frac {d}{dt}\left(\frac{\partial \cos(\theta - \phi)\dot \theta \dot\phi}{\partial \dot\theta}\right) $$ Why the correct result is $$\cos(\theta-\phi)\ddot \phi+\sin(\theta-\phi)\dot…
sunrise
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What will be the derivative $2 (\ln (x))^{x/2}$?

I'm not sure, that my steps are valid, so $ln(f(x))=ln \cdot (2 \cdot ln(x))^{x/2}=x^{2} \cdot ln \cdot(ln(x))$ so I get $\frac{1}{f(x)} \cdot f'(x)$ and I have to multiply both sides with $f(x)$, am I right?
Zauberkerl
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From the given information I have to find $f'(0)$

It is given that a function $f$ is differentiable everywhere, and $f(0)=5$. If $f(x)<5$ for all nonzero $x$ then what is the value of $f'(0)$?. Now I see that $0$ is a point of maximum of the function, which is differentiable at $0$, which means…
Not Euler
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Find $F'(x)$ given $ \int_x^{x+2} (4t+1) \ \mathrm{dt}$

Given the problem find $F'(x)$: $$ \int_x^{x+2} (4t+1) \ \mathrm{dt}$$ I just feel stuck and don't know where to go with this, we learned the second fundamental theorem of calculus today but i don't know where to plug it in. What i did: chain rule…
Need4Sleep
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What's the derivative wrt. $x$ of $\int_0^x f(x-s)\,ds$?

I know how to compute the derivative of $f(x-s)$ and by the fundamental theorem of calculus the derivative of $\int_0^x f(s)\,ds$ is $f(x)$. But I can't figure out how to do it when they're mashed together as in $\int_0^x f(x-s)\,ds$. The presence…
Sebastian Oberhoff
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Solving derivative of squared error where the predictor is a sigmoid function

$\newcommand{\sigmoid}{\operatorname{sigmoid}}$In the book "Make your own neural network" by Tariq Rashid, I have to take the derivative of my cost function which is: $$ \left(t-\sigmoid\left(\sum_j w_{jk}\times o_j\right)\right)^2 $$ where $t$ is…
Jesse
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Derivative Solution

I understand the first step but at the second step how do they come to negative one in the numerator? Also why do they show the definition of derivative again for step 2? How does this produce a negative one?
Jinzu
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$\inf\limits_{x \in [a,b]}|f^{\prime} (x)| \geq \frac{1}{b-a}$

Let $f \in C^1([0,1])$ be a non-decrease function such that $f(0)=0, f(1)=1$. Does there exist $[a,b] \subset [0,1]$ such that $\inf\limits_{x \in [a,b]}|f^{\prime} (x)| \geq \frac{1}{b-a}$?
user503375
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Two points on curve that have common tangent line

Find the two points on the curve $y=x^4-2x^2-x$ that have a common tangent line. My solution: Suppose that these two point are $(p,f(p))$ and $(q,f(q))$ providing that $p \neq q$. Since they have a common tangent line then: $y'(p)=y'(q),$ i.e.…
RFZ
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Can you have a vertical tangent where a function is undefined?

Can you have a vertical tangent where a function is undefined? For example, the function $y = 1/x$ is undefined at $x=0$, but that's where the denominator of its derivative is equal to $0$.
user245640
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Longest derivative "loop"?

If we keep differentiating $\sin x$ we eventually arrive back at $\sin x$: $$ \begin{align} y &= \sin x \\ \frac{dy}{dx} &= \cos x \\ \frac{d^2y}{dx^2} &= -\sin x \\ \frac{d^3y}{dx^3} &= -\cos x \\ \frac{d^4y}{dx^4} &= \sin x \end{align} $$ It has…
minseong
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Why do derivatives behave like fractions when they are not fractions but an operator as a whole?

The operator $\frac{d}{dx}$ sometimes behaves like a fraction.For eg. when we use chain rule to write $\frac {d}{dy}=\frac {d}{dx}\frac {dx}{dy}$ it apparently seems the $dx$ cancel out to give the previous thing.Also when differentiating…
Soham
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