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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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How to recognize, where function has no derivative?

For example this function: $f(x) = (x + 1)|x + 3| + 2$ it has no derivative in $x = -3$, but how can I discover it?
user50222
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Twice-differentiability of a function

I have $\sigma(s) \in C^0[0,\infty)$, non-decreasing with $\sigma(0)=0$. Define for $t>0$, $$ \tau(t) = \frac{1}{t^2} \int_{0}^{2t} \int_{0}^{2r} \sigma(s)ds dr $$ with $\tau(0)=0$. Is $\tau(t)$ necessarily twice-differentiable and non-decreasing…
Serikbolsyn Duisembay
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The definition of vector derivatives

I am reading a textbook about the derivatives, and it says.. Let $x, y, f(x)\in \mathbb{R}$ then we can define $f'(x)$ as the following: $$\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{h}=f'(x)$$ However if $x$ is a vector, then equivalent to the above…
Denny
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How to go from $dx = -t^2dt$ to $\frac{d}{dx} = -t^2\frac{d}{dt}$

Suppose $x = 1/t$. So now $x$ is a function of $t$, i.e., $x(t)$. So $$\frac{dx(t)}{dt} = -t^{-2} \Rightarrow dx(t) = -t^{-2}dt$$ This problem is from the textbook: advanced mathematical methods for scientists and engineers How to go from "$dx =…
Denny
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Let $f$ be differentiable on $[a,b]$, $f(a)=0$ and there is $c\geq 0$ such that $|f'(x)|\leq c|f(x)|$ then $f(x)=0$

Prove that if $f$ be differentiable on $[a,b]$, $f(a)=0$ and there is a constant $c\geq 0$ such that $|f'(x)|\leq c|f(x)|$ for all $x\in [a, b]$, then $f(x)=0$. Any piece of advice would be much appreciated.
user62498
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Two derivatives of $y=\arcsin 2x\sqrt {1-x^2}$?

$y=\arcsin 2x\sqrt {1-x^2} ,\frac{-1}{\sqrt2}\lt x \lt \frac{1}{\sqrt2}$ I. Substituting $x=\sin m$ $y=\arcsin (2\sin m \cos m)=\arcsin (\sin2m)=2m=2\arcsin x$ $\Rightarrow y'=\frac{2}{\sqrt{1-x^2}}$ II. If you substitute $x=\cos m$, you get $y=2m=2…
Raknos13
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Derivative in 1D as a linear transformation with reminder

There are many topics with the derivative definition, but I couldn't find a precise answer to my doubts. In one of the formulation the derivative of a function in a given point $x_0$ is a number $a\in\mathbb{R}$ such as: $$f(x_0+h)=f(x_0) + a\cdot h…
rk85
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Find The value of $a,b,c$.

If $f(x)=a|\sin x|+be^{|x|}+c|x|^3$ is differentiable at $x=0$, find the values of $a,b,c$. I know that the derivative exists at $x=0$ iff $f'(0^+)=f'(0^-)$, but I can't find $f'(x)$. Please help. Thanks in advance.
Argha
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If $f(0)=0$, can I say that $f'(0)=0$ as well?

Just a simple question on derivatives: If $f(0)=0$, then can I say that $f'(0)=0$ or it is completely wrong? But, if $f(x)=0$, then can I say that $f'(x)=0$ or and this is completely wrong?
Leos Kotrop
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How to find $\frac{d^n}{dx^n} e^x\cos x$

How can I get a formula for the n-th derivative of this function? I know that it cycles every 4 derivatives with a factor of $-4$. $e^x(\cos x-\sin x) \to e^x(-2\sin x) \to -2e^x(\sin x+\cos x) \to -4e^x\cos x$
Семён Юрченко
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Show that $ \exists \ c \in (a,b) \ $ such that $ \ f(b)-f(a)=cf'(c) \ln (\frac{b}{a}) \ $

Let $ \ f \ $ be a function continuous on $ \ [a,b] \ $ and differentiable on $ \ (a,b) \ \ and \ \ 0
MAS
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derivative caculation

I met a problem that described as follow. I am not sure the title is suitable for my problem. If you have any advices about the title or the description, please comment below. Aready…
Nick Dong
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Where f'(x) is defined if $f(x)= \left|x-1\right|$

i solved for f'(x) and i got $\frac{x-1}{\left|x-1\right|}$. i think that this is defined for 1>x>1. as $\frac{x-1}{\left|x-1\right|}$ can't be defined for x=1 because the denominator will be 0. I don't know if this is correct
john.r
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Are $\left(\frac{dx}{dt}\right)^2$ and $\frac{d^2 x}{dt^2}$ same things?

Is the second derivative of x(t) with respect to t equal to the square of the first derivative of x(t) with respect to t? In other words is the following correct: $$\frac{d^2x}{dt^2} = \left(\frac{dx}{dt}\right)^2 =…
user1999
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Differentiate then re arrange to find a value

Question: The height of a body is described by: $h = \frac{\ln\left(t^2-B\right)}{a}$ where is the time in seconds and and are constants. a) Write in terms of , and ℎ. b) Determine $\frac{ℎ}{}$ c) Given that $\alpha = 0.2$ and $ = 9$,…
PMA
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