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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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Differentiation of a Square root

Can the following be differentiated using first principles or the power rule? $$y = (x^3+6x^2+3x-10)^{0.5}$$ I know I have to get it into an expression where each term on the right hand side has its own power, instead of the whole thing having a…
GoodChessPlayer
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$\frac{d}{dy} F(g(y),y) = ?$

Given that we know if the integral of $f(y)$ is $F(y)$ then we can say that $\frac{d}{dy} F(y) = F'(y) = f(y)$. But what does $\frac{d}{dy} F(g(y),y)$ equal to? Can we say that it is $f(g(y),y)?$
Naz
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Differentiability of a function and its square root

Consider a function $f:\Theta \subseteq \mathbb{R}^l \rightarrow [0,\infty) $. Let (1) $\sqrt{f(\theta)}$ is differentiable at $\theta_0 \in \Theta$ (2) $f(\theta)$ is differentiable at $\theta_0\in \Theta$. Question: Are (1) and (2) equivalent,…
Star
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Differentiate without know formula $[\arcsin x]' = \frac{1}{\sqrt{1-x^2}}$

Is there any way how to get differentiate of $\arcsin x$ without memorize it? $$[\arcsin x]' = \frac{1}{\sqrt{1-x^2}}$$
DavidM
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Understanding meaning of $f''$ for $x^2$ and $x^4$

If $y=x^4$ Then $y' = 4x^3$ and $y'' = 12x^2$ At $x=0$, $y'=0$ and $y''=0$. So, at $x=0$, the gradient is zero is not increasing or decreasing at that point. I can believe this if I look at a plot of $y=x^4$. Now, $y=x^2$ has a similar shape, so we…
User 17670
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Calculate the value $-te^{-t} - e^{-t}$ at t $\rightarrow\infty$

I am trying to evaluate the following equation at $t \rightarrow \infty$ and $t \rightarrow 0$: $-te^{-t} - e^{-t}$ I am trying to use L'Hopital's Rule to evaluate $-te^{-t}$ at $\infty$, so it becomes $\frac{e^{-t}}{-t^{-1}}$, but it does not…
Joseph
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Can there be a function, that with the slope $f'(x_0) = 0$ at the single root $x_0$

Let $f$ be a function of the form $$ f(x) = \sum_{i = 0}^n a_i x^i = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x+ a_0 $$ with $a_n, a_{n - 1}, ..., a_1,a_0 \in \mathbb{R}$. Let $x_o$ be a single root of $f$ $$f(x_0) = 0 \iff f(x) = (x - x_0) * p(x)$$ Is…
Entimon
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Steps of finding an absolute extremum on an open interval

$$f(x)=\cot x-\sqrt 2 \csc x,\quad I=(0,\pi)$$ Show that the function $f$ has an absolute extremum on the given interval $I$ and find that value. I've found the local maximum point from the first derivative. I've showed that the second derivative at…
owlswollen
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construct a function with dense critical value

I am learning derivative of a function from $\mathbb R$ to $\mathbb R$. If $f$ has zero derivative at $x$, we call $x$ is a critical point of $f$, and $f(x)$ is critical value. If critical points $C(f)$ is dense in $\mathbb R$, can we say $f$ is a…
gaoxinge
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A function $g \in C^{1} (\mathbb{R})$ with $g(0)=0$ verifies $|g(s)| \leq C|s|$ for $s \in [-M,M]$

I need to show that a function $g \in C^{1} (\mathbb{R})$ with $g(0)=0$ verifies $|g(s)| \leq C|s|$ for $s \in [-M,M]$. This is what I have done: Using Mean Value Theorem, we have: $$\frac{g(s)-g(0)}{s-0}=g'(c)$$ For some $c \in {[-M,M]}$. Therefore…
D1X
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how to find the slope of a curve at origin

What is the slope of the curve $x^3 + y^3 = 3axy$ at origin and how to find it because after following the process of implicit differentiation and plugging in $x=0$ and $y=0$ in the derivative we get $\frac{0}{0}$.....how to solve these kinds of…
shashank
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How do I calculate the derivativee of the function $x\mapsto \sqrt{x}$ in two ways?

I need to calculate the derivative of the following function in two ways: $$ f\colon\mathbb{R}^+\to\mathbb{R}^+, \quad x\mapsto \sqrt{x} $$ a) by means of differential quotient b) using the derivation rules for powers I need to determine the…
Rahul
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What does $f^{(90)}(0) = -\frac{90!}{18!}$ really mean?

Given $f(x) = \cos(x^5)$ And that $f'(0) = 0$ What does $f^{(90)}(0) = -\dfrac{90!}{18!}$ really tell us? (Note, my previous question solved for this value). How can the rate of change of $f$ at $0$ be $0$, but the rate of change of the range of…
user17753
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Differentiating $x^2=\frac{x+y}{x-y}$

Differentiate: $$x^2=\frac{x+y}{x-y}$$ Preferring to avoid the quotient rule, I take away the fraction: $$x^2=(x+y)(x-y)^{-1}$$ Then: $$2x=(1+y')(x-y)^{-1}-(1-y')(x+y)(x-y)^{-2}$$ If I were to multiply the entire equation by $(x-y)^2$ then continue,…
john2546
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