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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.

33197 questions
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Reverse mode differentiation vs. forward mode differentiation - where are the benefits?

According to Wikipedia forward mode differentiation is preferred when $f: \mathbb{R}^n \mapsto \mathbb{R}^m$, m >> n. I cannot see any computational benefits. Let us take simple example: $f(x,y) = sin(xy)$. We can visualize it as graph with four…
user1700890
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Differentiable but not continuously differentiable.

Given $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as $$f(x)=\left\{\begin{array}{cc}x^2\sin\left(\frac{1}{x}\right)&,x\neq 0\\ 0&,x=0\end{array}\right\}.$$ I am trying to prove $f$ is differentiable at $x=0$ but not continuously differentiable…
Jennifer
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Derivative is just speed of change?

In school we've been told that derivative of $x^2$ is $2x$. Also I've read that derivative is simply a speed of value change. So if $$f(x)=x^2$$ then, using simple explanation, derivative of that function would be $$f'(x)=f(x+1)-f(x).$$ Now if we…
MeelStorm
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Differentiation of vector norms

I want to solve the following equation $$\frac{\partial}{\partial {\bf \beta}} \left[||{\bf y}-{\bf X}{\bf \beta}||^2 + ||{\bf \beta}||^2\right] = 0$$ for $\beta$. Here ${\bf y}$ and ${\bf \beta}$ are vectors and ${\bf X}$ is a matrix. I am having…
user61300
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If a derivative of a continuous function has a limit, must it agree with that limit?

Suppose we have a continuous function $f : \mathbb{R} \to \mathbb{R}$. Suppose also that for a certain point $c$, $\lim_{x \to c} f'(x)$ exists. Must $f'(c)$ exist as well, and be equal to this limit? This isn't quite the same as asking if…
Tanner Swett
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Zeroes of derivatives of high order

The problem is following. Let $f:(-1,1)\to [-1,1]$ has $n$ derivatives. Prove that there exists a number $\alpha_n$ (independent from $f$) such that condition $|f'(0)|\geq \alpha_n$ implies that equation $f^{(n)}(t)=0$ has at least $n-1$ distinct…
Georgii Riabov
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What is the derivative of max and min functions?

If I define a function: $f(x) = \max[g(x),h(x)]$ What is $f'(x)$?
Jonathas Maciel
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Showing $f'(x) = f(x)$ implies an exponential function

Possible Duplicate: Proof that $\exp(x)$ is the only function for which $f(x) = f'(x)$ How can I show the statement $f'(x) = f(x)$ implies the function is defined as $f: \mathbb{R} \rightarrow \mathbb{R} : x \rightarrow a\cdot \exp(x)$ without…
billcarson
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Is $x^{1/3}$ differentiable at $0$?

It occurred to me that functions that are "smooth" but have "infinite slope" may not be considered differentiable at the points where their slope is infinite. An easy example of this is $x^{1/3}$, which is "smooth" in a visual sense (i.e. there are…
user3002473
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Derivatives 101: what does "with respect to" mean?

I'm studying derivatives 101 and I can't get my head around the phrasing "with respect to" something. Eg in chain rule we calculate the derivative of outer function with respect to inner + derivative of inner with respect to x. But what does it…
ilmoi
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If $f$ is bounded and twice differentiable in $\mathbb{R}$, show that there exists $\xi\in\mathbb{R}$, s.t. $f''(\xi)=0$.

My idea: If $f$ has maximum and minimum, then $f'=0$ at these two points, and the conclusion is further derived using Mean Value Theorem. But what if $f$ has no maximum/minimum, like $f=\frac{1}{1+e^{-x}}$? $f$ is twice differentiable in…
useprxf
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Total Derivatives and Total Differential

I am confused between total derivatives and total differential. What is the difference between total derivatives and total differential?
AB408
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Derivative of infinite sum

I was thinking about derivative of infinite sum of functions, i.e. $$f(x) = \sum_{i = 0}^\infty g_i(x)$$ $g(x)$ is continuous in domain of $f$ Because if $(f+g)'(x) = f'(x) + g'(x)$ then $\left(\sum\limits_{i = 0}^{\infty} g_i(x)\right)' =…
Hauleth
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derivative of indicator function

I have an indicator function $I(D\leq Q)$which is equal to $1$ if $D\leq Q$ and $0$ otherwise. What would be derivative of this function with respect to different variables such as $D$ or $Q$ or $P$ ($D$ is a function of $P$). Clarification to what…
Eln
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Is $\max(0, x)$ a differentiable function?

It appears that $\max(x, y)$ isn't differentiable according to this question. However, the explanation is due to the fact that $\max(x, -x) = \lvert x\rvert$, and since there won't be the case $\max(0, -0)$, does this mean that this function is…
Ren
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