Mathematics Stack Exchange
Mathematics Stack Exchange

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
2
votes
2 answers

Two methods of finding out $\frac{d^2y}{dx^2}$ when $\frac{dy}{dt} = 4t-2$ and $\frac{dx}{dt} = 2t + 3$

Find $\frac{d^2y}{dx^2}$ given that $\frac{dy}{dt} = 4t-2$ and $\frac{dx}{dt} = 2t + 3$ at $t=2$ I tried two methods for this question, both of which give me different answers. Which one is wrong and why so? Method $1$: $\frac{dy}{dt} = 4t-2$ hence…
marks_404
  • 631
2
votes
1 answer

Is an infinite product of functions differentiable if it's components are?

$$ p(\theta) = \left[\lim_{t \to0} p(t) \right] \lim_{n \to \infty}\prod_{i=0}^{n}( \frac{\cos^2 \frac{ \theta}{2^i} +1}{2})$$ I know each term in the product is differentiable but does that mean the total product will be ? I thought of doing a…
tryst with freedom
  • 11,538
2
votes
2 answers

What functions satisfy $D^* f = Df$?

I came across one exercise that I found pretty interesting. The problem gave us a new definition of derivate $D^*f(x) = \lim_{h \to 0} \frac{f^2(x + h) - f^2(x)}{h}$. It asks us for what functions does $D^*f = Df$. I know that $D^*f = 2fDf$ but I am…
Dachuan Huang
  • 151
2
votes
2 answers

Derive a new equation from $m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$

$$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$ $$\implies m^2=\frac{m_0^2}{1-\frac{v^2}{c^2}}$$ $$\implies m^2c^2-m^2v^2=m_0^2c^2$$ Differentiating the equation, $$2m \;dm\;c^2-2m\;dm\;v^2-2v\;dv\;m_0^2=0$$ Our book says when we differentiate…
user876873
2
votes
0 answers

Derivative Product rule for non-commutative objects?

lets say I have objects $f$ and $g$, for which one can define a derivative with the typical properties. The product rule would be expected to be $$ d(fg)=(df)g+f\,dg $$ But what if $f$ and $g$ are not commutative? $$ [f,g]\neq 0\implies…
Anon21
  • 2,581
2
votes
1 answer

Effects of variables' change on total change in a product function

Prelude My topic is related to existing posts found here and here; although, my questions are different. My post explains my objective and what has and has not worked for me. I have met my objective and provided my solution. My questions pertain to…
AWaddington
  • 171
2
votes
1 answer

Common notation of differentials

I have a doubt which confuses me a lot. If f(x) is a function in x then does f’(2x) mean $d f(2x)/d 2x $ or $d f(2x)/ d x $ as when we integrate it the result is f(2x)/2 and the same for $ f’(x^2)$. Is it $d f(x^2)/d x^2 $ or $d f(x^2)/d x $ and…
Nil
  • 127
2
votes
2 answers

Computing the gradient of Cross Entropy Loss

The categorical cross entropy loss is expressed as: $$L(y,t) = -\sum_{k=1}^{K}t_k\ln{y_k}$$ where $t$ is a one-hot encoded vector. $y_k$ is the softmax function defined as: $$y_k = \frac{e^{z_k}}{\sum_{j=1}^{K}e^{z_j}}$$ I want to compute the…
mathnewbie
  • 47
2
votes
1 answer

Intuition behind stolz caesaro theorem

The stolz caesaro theorem seems to be a discrete analogue of L'hopitals rule. I can understand l'hopitals rule via the taylor series, that is: $$ \lim_{x \to a} \frac{P(x)}{Q(x)} = \frac{ P(a) + \frac{dP}{dx}|_a (x-a) + O ( (x-a)^2) }{ Q(a) +…
tryst with freedom
  • 11,538
2
votes
2 answers

simplify fraction (stone drops off a cliff)

How do I get from $$ \frac {dt} {T} = \frac {dx} {gt} \sqrt { \frac {g} {2h} } $$ to $$ \frac {dt} {T} = \frac {1} {2 \sqrt {hx} } dx $$ where $x(t) = \frac {1} {2} gt^2 $ and $ T = \sqrt { \frac {2h} {g}}$. I'm currently struggling with Griffiths'…
Philip Witt
  • 33
2
votes
1 answer

if $F'(ax) = G(x)$, then $F'(nax) = nG(nx)$?

A question I found stated that: Given $$\frac{d}{dx}F(ax) = G(x)$$ then $$\frac{d}{dx}F(2ax) = \dots$$ with choices: 1.) G(2ax) 2.) G(ax) 3.) 2G(2x) 4.) 2G(x) I got the third option as the answer by assuming…
Clarence The Pro
  • 23
2
votes
2 answers

If $f$ is a function differentiable at $a$ find: $\underset{h\rightarrow 0}{\lim} \frac{f(a+7h)-f(a-9h^2)}{h}$

If $f$ is a function differentiable at $a$ find: $\underset{h\rightarrow 0}{\lim} \frac{f(a+7h)-f(a-9h^2)}{h}$ I am struggling to understand what to do. I tried brute forcing this question and I get $\frac{f(a)-f(a)}{0}$ which makes no sense to me.…
Nuemann12
  • 71
2
votes
1 answer

proving that $f'(x) > g'(x)$ implies $f$ and $g$ cannot intersect more than once.

I'm not completely certain if the statement is true, but intuitively I think it is. My sort of intuitive explanation would be if there is a point where $f$ and $g$ are equal, then the fact that $f'$ is greater than $g'$ would prevent them from…
jazhang
  • 257
2
votes
3 answers

Derivative of a function of the form $f=f(x,g(x))$

Consider a function of the following form $f=f(x,g(x))$. For the derivative of this function, we do the following. $$ \frac{df}{dx} = \frac{df}{dx} \bigg\rvert_{g}+\frac{df}{dg} \frac{dg}{dx} $$ I have the following question about the above…
prananna
  • 21
1 2 3
…
99 100