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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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how to solve for the coefficients of a simple linear rational function

so what i want is to find the coefficients which is $a,b,c,d$ in terms of $f(x)$ where $f(x)=(ax+b)/(cx+d)$ only in terms of $f(u)$,$f'(u)$ and $f''(u)$ where $u$ is some constant i solved for a similar question but its instead…
Ashton Smith
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Prove that there is a strict extremum in $x_0$.

Let $f: (a, b) \to \mathbb{R}$ and $f$ differentiable in $x_0 \in (a, b)$ and $f'(x_0) \ \neq 0$ I want to prove that $g(x) = (f(x) - f(x_0))(x - x_0)$ has in $x_0$ severe extremum. Looks like $f$ is monotonous and we can prove in this case. But…
Someone
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Question on derivative

I want to differentiate $H(p(t),q(t))=1 $ with respect to $t$, where $H:\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} $ is a convex function. I think that it is: $\displaystyle \frac{dp}{dt} \frac{\partial H(p,q)}{\partial p}+ \frac{dq}{dt}…
Vrouvrou
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What is the "average slope"?

Derivative This article says the following: To find the slope at the desired point, the choice of the second point needed to calculate the ratio represents a difficulty because, in general, the ratio will represent only an average slope between…
user366312
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F(x) is a function differentiable and monotonic strictly increasing on the OPEN interval (a,b).

$f(x)$ is a differentiable and strictly increasing function on the open interval $(a,b)$. Prove or disprove that $f'(c)>0\; \forall c\;\in (a,b)$. I was trying to apply Lagrange's derivative theorem, but the condition that the function must be…
Ben Reznik
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How to compute the derivative of $clip(\cdot)$ function?

I am wondering what the correct derivative of the $clip(x,min\_value, max\_value)$ function is, where the $clip(\cdot)$ function clips $x$ to the range $[min\_value, max\_value]$. For convenience, take $f(\cdot) = clip(\cdot)$, $a = min\_value$, and…
Daniel B.
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Derivative equal to original function

How many different functions $f(x)$ exist such that $f'(x) = f(x)$? The ones I know of right now are $f(x) = 0$ and $f(x) = ne^x$, for any real number $n$. What other functions satisfy this property?
thesilican
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Differentiability of a function with absolute value and sign function

I have the following function: $$|\sin{x}|\cdot sgn{x}$$ where $$ -\pi
john doe
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Under what conditions is $\frac{dF(x)}{dx}\Big|_{x-x_0} = \frac{dF(x-x_0)}{dx}\Big |_x$?

I have a very complicated function for which I need to evaluate $$ \frac{dF(x)}{dx}\Bigg|_{x-x_0}.$$ How can I tell when this will be equivalent to evaluating $$ \frac{dF(x-x_0)}{dx}\Bigg|_x ?$$ Under what conditions on $F$ is this equivalent? Would…
kevinkayaks
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dy/dx treated as fraction?

My question might seem very dumb but here goes. Say we have y = f(x). If we have g = dy/dx, we can also say that gdx = dy, and this works. Why does it work? I know that dy/dx isn't a fraction but what underlying properties are being implicitly used…
Kramen
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To show $f(x)=e^{x^2} +2x(xe^{x^2}-e^{-x^2})$ is positive in the interval $x\in[0,1]$

$f(x)=e^{x^2} +2x(xe^{x^2}-e^{-x^2})$ and $x\in[0,1]$ I want to know how $f(x)$ is positive in the interval. $e^{x^2}$ is +ve but $(xe^{x^2}-e^{-x^2})$ is not always +ve. So how can I show that $e^{x^2}$ is always greater than…
Rajesh Marndi
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Complicated derivative question from a calc 2 student

Question from a Calc 2 student. To solve the following derivative $$\frac{d}{dn}\left(\frac{1}{2^{2n+1}}\frac{\left(\frac{1}{2}\right)^{2n+2}}{\left(2n+1\right)\left(2n+2\right)}\right)$$ can a dummy variable be used for $2n+2$ or $2n+1$ to make…
findingmyway
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Find the derivative of the following

Can someone help me solve this?. I have hard time understanding the lesson, and our teacher will give us a quiz next meeting. She leave this as our exercise. I just want to see the answer and solution. I can't comprehend of what she had been saying…
Suan Suan
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Is it possible to plot a function with a vertical tangent line while the plot of the function has no vertical line segment?

Is it possible to plot a function with a vertical tangent line while the plot of the function has no vertical line segment?
Christina Daniel
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If $f$ is a function differentiable at $a$ find: $\underset{h\rightarrow 0}{\lim} \frac{f(a-h^2)-f(a)}{h}$

If $f$ is a function differentiable at $a$ find: $\underset{h\rightarrow 0}{\lim} \frac{f(a-h^2)-f(a)}{h}$ I figure that the answer is $\infty$, but I a torn on whether I am correct. Any idea whether I am correct or if I have an issue. My work is…
Nuemann12
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