Mathematics Stack Exchange
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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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Can we prove if a/b is greater than da/db

I was reading Keynesian Economics and came across this relation: $$\frac CY > \frac {dC}{dY}$$ provided that $$Y = C+S$$ where Y is total income of an individual, C is Consumption of the individual and S is Saving of that individual. Basically the…
ATK
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What is the derivative of $f(x) = (x − y)^2 g(x)$

I'm studying about Newtons Method and my notes say that if $f(x) = (x − y)^2 g(x)$ then $$f'(x) = 2(x-y)g(x) + (x − y)^2 g'(x)$$ Is this true? I thought it would be $$f'(x) = 2(x − y) g(x)$$ because $g(x)$ is just part of the multiplication.
user2272600
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Suppose $f(0)=0$, and $f'(x) < \frac{1}{2}$. Show $f(4) < 2$

Problem: Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be differentiable. Suppose $f(0)=0$, and $f'(x) < \frac{1}{2}$ for all $a$. Show $f(4) < 2$. This question is intuitively pretty simple, but I'm not sure what is generally regarded as sufficiently…
Snowball
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How to calculate $\frac{\partial \mathrm{logdet}(I+XX^T)}{\partial X}$?

How to calculate $\frac{\partial \mathrm{logdet}(I+XX^T)}{\partial X}$? ($X\in\mathbb{R}^{n\times m}$) I know that $\frac{\partial \mathrm{logdet}(I+X)}{\partial X}=(I+X)^{-T}$, can we use it?
Lee
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Proving if $f$ is differentiable at 0 when f(0) = 0

Suppose that we have a function $f$ that $f(0)=0$ and for any $x$, $$|f(x)|>=\sqrt{|x|}$$ How could we prove that $f(x)$ is differentiable at zero or not? P. S. My first attempt was to prove something like a discontinuity for $f$ at zero that not…
Jigsaw
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Biggest directional coefficient

How do you get the greatest directional coefficient's exact points if you have the equation of the curve? This is not a mathematics problem. I have a mathematics problem, where I must count out at which points there is the greatest directional…
Sara
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Why is 2y * dy/dx = 1 when y = sqrt of x

I saw this as one of the answers to $y=\sqrt{x}$ but don't get the $2y$ part \begin{align} y&=\sqrt{x}\\ y^2&=x\\ 2y\frac{\mathrm{d}y}{\mathrm{d}x}&=1\\ \frac{\mathrm{d}y}{\mathrm{d}x}&=\frac{1}{2y}\\ &=\frac{1}{2\sqrt{x}}\tag{2} \end{align}
nicevan
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Find increasing and decreasing intervals / critical points

Here's my equation: $y=x(x-4)^{3}$ I'm supposed to find the increasing and decreasing intervals, which I know how to do for other problems but this one is giving me issues. My first question is: should I expand the equation before taking the…
Grav
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How can we define the rate of change of a function at a single point?

I am studying derivatives and the basic idea of derivatives is to define the rate of change of a function. Now change in mathematics occurs in between two points so how can we even define the rate of change at a single point. It's completely invalid…
Ritanshu Singh
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Differentiation Properties Question

So I am running into this question, and I am not sure if desmos is wrong in this case. The question is true or false: If f · g and g is differentiable at x = a, then f is differentiable at x = a. A counter example would be: g(x) = 0 f(x) =…
user722457
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Differentiability definition

I am trying to understand the definition of differentiation, I am working on some problems and ran across this one: If f+g is differentiable at x = a and f is differentiable at x = a, then g is differentiable at x = a I think this is true…
user722457
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Are there any inflection points?

$$ F(x) = \begin{cases} x^2 & x \le 0 \\ 0 & 0 \le x \le 3 \\ -(x-3)^2 & x>3 \end{cases} $$ My question is does this function have any points of inflection? Double Derivative at $x=0,3$.Thus the necessary condition is satisfied for all points such…
ghghghghghg
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Given $x^y=e^{x-y}$, find $\dfrac{dy}{dx}$.

I have done the sum by taking log but not getting my final answer right. Please help me
L Lawlit
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How can $\mathbf r \times \mathbf (\frac{d}{dt}(mv))=\frac{d}{dt}(\mathbf r \times \mathbf mv)$? and other derivatives

Do you see why: $m\int \frac{dv}{dt} v dt = \frac{m}{2}\int \frac{d}{dt}(v)^2 dt = $ How can you put the $v$ inside the differential $\frac{d}{dt}$, if it is time-dependent? And where does the $\frac{1}{2}$ come from? Another (easy) one: $\mathbf r…
user3203929
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Does 'derivative exists at a point' imply 'derivative is continuous at the point'?

The condition of a derivative existing at a point is that slopes to the curve drawn from the left hand side of the point, and the slopes drawn from right hand side of the point must approach the same value. The condition for continuity at a point is…
Ryder Rude
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