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.

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general formulation for 1/g(x) derivative

Is there a general formulation for $\frac {d^n(g(x)^{-1})}{dx^n}$ ? Something like $$\frac {d^n(g(x)^{-1})}{dx^n} = \sum_{i=1}^{f(n)}\prod_{k=1}^{h(n)} ...$$
Nuno Silveira
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Logaritmic derivation!

Why can't I derive this function using normal methods?Text book says that i have to use something called "logarithmic derivation". I don't know if this term exists in English, but that is the direct translation from my language. Apparently you use…
user118751
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Finding the derivative of: $ y(x)=x⋅\sin(x)+x^2⋅\cos(3x) $

I am a student and I have a problem in solving this derivative, so please help me. The problem: $$ y(x)=x⋅\sin(x)+x^2⋅\cos(3x)$$ I give this problem to online derivative calculator like derivative-calculator.net but the answer is not equal to…
reza
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find the derivative $\displaystyle \frac{d}{dx}\int^a_x \tan(\tan(t))\,dt =$

Find the derivative: $\displaystyle \frac{d}{dx}\int^a_x \tan(\tan(t))\,dt = $ I tried to take the derivative but I am getting the wrong answer every time.
shikorina
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Question about continuous differentiability

Consider the function $f(x,y) = \frac{x}{1+\sqrt{x^2+y^2}}$. Its derivative with respect to $x$ can be calculated to be $\frac{1 + \frac{y^2}{\sqrt{x^2 + y^2}}}{1 + x^2 + y^2 + 2 \sqrt{x^2 + y^2}}$. Is it correct to say that $\frac{\partial f(x,y)}…
Sunny88
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Derivatives of second order

Consider a real function $f$ of one variable. Suppose the second order derivative exists. To find the second order derivative of $f$, I usually derivate $f$ two times. I start with $f$, and derivate to get the function $\frac{d}{dx} f(x)$ of one…
Elias
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Finding derivative using product and chain rule

I need to find first derivative of $x\sqrt{2-x^2}$. My approach Using product rule: $(2-x^2)^{1/2} + x\frac{\operatorname{d}(2-x^2)^{1/2}}{\operatorname{d}x}$ Using chain rule: $(2-x^2)^{1/2} + x\left[\frac{1}{2}(2-x^2)^{-1/2}…
J.Olufsen
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Derivative of a fraction with respect to another

I've found this derivative on a textbook $\dfrac{d(c_{t+1}/c_t)}{d(\dfrac{\gamma}{c_t}/\dfrac{1-\gamma}{c_{t+1}})}=\dfrac{1-\gamma}{\gamma} \dfrac{d(c_{t+1}/c_t)}{d(c_{t+1}/c_t)}=\dfrac{1-\gamma}{\gamma}$ I would like to understand the first…
Luigi
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How do you differentiate -|t|?

How do you differentiate $-|t|$? Using Wolframalpha it says to re-write $-|t|$ as ($-\sqrt{t^2}$). Why? (This is part of a bigger question, that being to calculate the differential of $e^{-|t|/T}$ ).
Polly
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Determine if a function is a total derivative

Lagrangian is defined up to addition of a total derivative of function of positions and time. Now suppose we have a function $f(x,\dot x,t)$. How can one show (check) that $$\not\exists g(x,t):\; f(x,\dot x,t)=\frac{\text{d}}{\text{d}t}g(x,t)$$ ?
Ruslan
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Find the second derivative of the given function

If $$x=a(\cos \theta + \theta \sin \theta) $$$$ y=a(\sin \theta- \theta \cos \theta) $$ prove that $$\frac{d^2y}{dx^2}= \frac{\sec^3 \theta}{a \theta}$$ Can you solve this for me? I tried finding $\frac{dy}{dx} $ by dividing $\frac{dy}{dt} $ by…
chndn
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Solving a differential equation with composite functions

If $c=f(a+e^b)+g(a-e^b)$ where $f$ and $g$ are functions of $a+b^2$ and $a-b^2$ respectively, find $c$ such that when $b=0$, you find that $c=0$ and $\frac{\partial c}{\partial b}=1+a$.
Jay Christian
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Differentiation: what rule am I missing here?

I am trying to differentiate the below with respect to c: $\left(\frac{a-b}{c-b}\right)^d$, however I get an answer different to what Mathematica (and other sources) is telling me, which…
coconut
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Differentiate $x \sqrt{1+y}+y \sqrt{1+x}=0$

If $x \sqrt{1+y}+y \sqrt{1+x}=0$, prove that $(1+x^2)\frac{dy}{dx}+1=0.$ The answer I got is $$\frac{dy}{dx}= -\frac{2 \sqrt{1+x} \sqrt{1+y}+y}{x+2 \sqrt{1+x}\sqrt{1+y}}$$ but I cannot simplify it further. Please provide your assistance.
chndn
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Does differentiation change the units of measurement in mathematical equations?

While i'm doing the math homework, I find something very strange. I am confused by a textbook's answer. The Question The textbook's answer So, Does differentiation change the units of measurement in mathematical equations?
Nie Philip
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