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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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Find the value of dy/dx when x=1

I know this is probably a really easy question. However I keep reading the examples again and again and constantly cant get the correct answer. Find the value of $\frac{dy}{dx}$ when $x=1$ $$ y = a^2x -ax^2$$ Is is possible someone could do a…
allquiet1984
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can a function be differentiable at a point if the difference quotient has a finite limit but the derivative has no limit?

In the case of x^2*sin((5x+4)/x). the difference quotient has limit 0 for x->0, but the derivative itself is 2x*sin((5x+4)/x)+ 4*cos((5x+4)/x), which has no limit for x->0. So is the function differentiable at 0?
MatSiv97
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Quotient rule of derivatives

Quotient rule of derivative is: $(\frac{f}{g})^{\prime}$ = $(\frac{f^{\prime}g - g^{\prime}f}{g^2})$ but when I compute a deriative of $\frac{1}{(1-x)}$ , it gives $\frac{1}{(1-x)^2}$ which is right but taking second derivative of this gives me…
Khan Saab
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Profit maximization through differentiation (cattle)

Each cow weighs 300kg. The cost of maintaining a cow is 20€ per day. The rate of change of the cows weight is 4kg a day. The market price is currently 50€ per kg and it is falling 50 cents a day. How much time should the cattleman wait to sell the…
BSD
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Is it meaningful to consider this derivative?

Consider some function $f(k) = \frac 1 {a_1 a_2 \cdots a_k}$ so that the number of terms in the denominator changes with $k$. My question is there anyway to discuss its derivative, if so, how would one go about beginning to differentiate…
Ryan Goulden
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Derivative - answer is not the same using two different routes

Good evening. I am so confused, I am trying to find the derivative of $x=e^t$ and $y=t^2e^{-t}$ to find the tangent of a parametric function. The only thing is, is that when I derive $dy/dt$ I get $-(t-2)te^{-t}$ and $dx/dt = e^t$; thus, $dy/dx =…
Gaby Allard
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A function that is not a derivative of any derivable function

That's basically it. I need to find a function on the interval $[0,1]$ that isn't a derivative of any derivable function. I've found one possible solution which sets $0$ for every $x \in \mathbb{Q}$, and $1$ for every $x$ from $\mathbb{R}\setminus…
waterlemon
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partial derivative of $x^{y^z}$

Can someone tell me the correct partial derivative of $x^{y^z}$ on the variable z. I got two solutions: $x^{y^z}\ln\left(x^y\right)$ and $x^{y^z}y^z\ln x\ln y$ They both seem right to me so I am confused. Which one is correct and why?
mandm
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Gradient Representations

Is $\frac{dy}{dx} \equiv \frac{\Delta y}{\Delta x}$ $\frac{\Delta y}{\Delta x}$ is often used in physics, for example: $I=\frac{\Delta Q}{\Delta t}$ I was just wondering, isn't this the same things as the derivative of $Q$ with respect to $t$?
Tobi
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Problem in solving a question related to derivative.

The question is : If $f : \mathbb R \longrightarrow \mathbb R$ is differentiable and bijective.Then is $f^{-1}$ differentiable? It is clear that here $f$ is either strictly increasing or strictly decreasing.But from here how can I proceed to prove…
Arnab Chattopadhyay.
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Showing that $f \in C^{\infty}(\mathbb{R},\mathbb{R})$

Let $f(x)=\begin{cases} e^{-(1/x)} & \text{for} \quad x > 0 \\ 0 & \text{for} \quad x \leq 0 \end{cases}$ Show that $f \in C^{\infty}(\mathbb{R},\mathbb{R})$ I need to show that $f(x)$ has derivatives of all orders at all points in $\mathbb{R}$. It…
B.Swan
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Operations with $\frac{dy}{dx}$.

While working on a problem I have found a solution. I am curious about a clean and correct way to write it down. I want to find the derivative of $y(x)=:y$. $$(5y^4+1)\frac{dy}{dx} + 1 = 0\\ \frac{dy}{dx} = -\frac{1}{5y^4+1}$$ Is it mathematically…
B.Swan
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How Square tin with side "6a" units after bending have length of a side of the square is "a" units?

Suppose by using square tin sheet with side "6a units", make a topless box of maximum volume by cutting equal squares at the corners and removing them and then bending the tin so as to form the sides of the box. What I don't understand is that how…
Zonnie
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Find all the points where $f$ has deviration on them

A function $f:[0,2] \to \mathbb R $ is given in this way : $f(x)=\inf\{|x-\frac{1}{n}|:n \in \mathbb Z^+\}$ How can one find all the points where $f$ has derivation on them? Note : I know that existence of the derivation is equivalent to the…
Tina Ashtiani
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How do you compute the derivative of $\int_x^0\frac{\cos(xt)}{t}dt$?

What is the derivative of $\int_x^0\frac{\cos(xt)}{t}dt$ with respect to $x$? Using Leibniz' rule, I think it equals $$ \begin{align} -\frac{\cos(x^2)}{x}+\int_x^0 -\sin(xt)dt &= -\frac{\cos(x^2)}{x}+\frac{\cos(xt)}{x}\bigg\vert_x^0 \\ &=…
Noomi Holloway
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