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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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Derivative by Definition of $\frac{\sin^2(x)}{e^x-1}$

I have to prove the derivative by definition of $$\frac{\sin^2(x)}{e^x-1}$$ $$f^{\prime}(x)=\lim_{\Delta x \to 0}{\frac{f(x+\Delta x)-f(x)}{\Delta x}}$$ $$\large f^{\prime}(x)=\lim_{\Delta x \to 0}{\frac{\frac{\sin^2(x+\Delta x)}{e^{(x+\Delta…
Antonio Velazquez Bustamante
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Understanding derivatives

I don't know if this is written somewhere else. I've looked all over the internet so apologies if this has already been covered. I'm doing Year 12 Maths in Australia for what it's worth. In our textbooks the formula to find the derivative of a…
mitch
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Two times differentiable function

We know that the two times differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ is such that $g(0) = 999$, $g'(0) = 1000$ and $|g''(x)| \le 10000$ for every $x \in \mathbb{R}$. Let $K := g(\frac{1}{1000})$. How to prove that $K$'s second…
malex
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Rolle's Theorem with roots

Let $f : [a, b] \to \Bbb R$ be $n$ times differentiable and have $n+1$ distinct roots (i.e. solutions of $f(x) = 0$) in $[a,b]$. Show that there is an $x \in [a, b]$ s. t. the $n^{\text{th}}$ derivative of $f$ has a root in $[a,b]$. I know that you…
Jacob Rodgers
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There is a function which is continuous but not differentiable

I have a function which is a convergent series: $$f(x) = \sin(x) + \frac{1}{10}\sin(10x) + \frac{1}{100}\sin(100x) + \cdots \frac{1}{10^n}\sin(10^nx)$$ This function is convergent because for any E you care to specify, the function has a term which…
user1928764
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Nth Derivative of a fucntion

Find the $N^{th}$ derivative of $$f(x) = \sqrt{\frac {1-x}{1+x}}$$ I have got $1^{st}$ derivative as: $\frac{-1}{(1-x)^{1/2}(1+x)^{3/2}}$ and $2^{nd}$ derivative as: $\frac{1-2x}{(1-x)^{3/2}(1+x)^{5/2}}$ and $3^{rd}$ derivative as:…
rndflas
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Prove that $f(c)=\frac12(c-a)(c-b)f''(\xi)$

A function $f:[a,b] \rightarrow \mathbb R$ is continuous on $[a,b]$ and $f''(x)$ exists $\forall x\in (a,b)$. If $a
Diya
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Calculate the derivative and find its domain: $\;f(x)= \sqrt{\ln(x)+2}$

I calculated the derivative as $$f'(x) = \frac{1}{2x \sqrt{2+\ln x}}$$ How do I find out the domain?
user3924310
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Equations of the tangents to the curve $f(x) = 2x^2 + 3$

Find the equations of the tangents to the curve $f(x) = 2x^2 + 3$ that pass through the point $(2,3)$. Include a sketch as part of your solution. First, I found the derivative of $f(x)$ which is $4x$, but I'm not sure what to do next. Thank you.
user148748
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Derivative of $f(x) = x^2 \sin(1/x)$ using the derivative definition

derivative of $f(x) = x^2 \sin(1/x)$ using the derivative definition When not using the derivative definition I get $\cos (1/x) + 2x \sin(1/x)$, which WolframAlpha agrees to. However when I try solving it using the derivative definition: $$\lim_…
Mollart
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Differentiation of the Law of Cosines, where a, b, c, A, B, and C are functions of time t

Is the differentiation of the law of cosines ($c^2= a^2 + b^2 - 2ab\cos C$) this? a, b, c, A, B, and C are functions of time t. $$2c \frac{dc}{dt} = 2a \frac{da}{dt} + 2b\frac{db}{dt} - 2b \cos C \frac{da}{dt} - 2 a \cos C \frac{db}{dt} + 2ab \sin C…
Josue
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Prove that $f(a) \leq f(x) \leq f(b) $

If the following data are given, prove that $f(a) \leq f(x) \leq f(b) $ f is differentiable on [a,b] and f'(x) $ \geq 0 \forall x \in (a,b) $ Is the following argument correct? $f'(x) \geq 0 \implies f $ is increasing on (a,b) $ \implies f(a) \leq…
S.Dan
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Finding the derivative of $\sqrt{x+\sqrt{x^2+5}}$

How to derive $y=\sqrt{x+\sqrt{x^2+5}}$ at $x=2$.I used logarithmic differentiation and chain rule over and over again but I can't get the right answer
user162143
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When can I say that $f(x) \gt g(x) \implies f'(x) \gt g'(x)$?

Are there cases when this relation holds? $$f(x) \gt g(x) \implies f'(x) \gt g'(x)$$ I.e. what are the conditions on $f(x)$ and $g(x)$ for that to be true? Is it even possible to determine them? In case it is always valid, how can it be proved?
rubik
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Lines tangent to parabola at point.

I'm struggling to figure out what I'm exactly required to do. The problem states "Compute which lines through the point $(1, 0)$ that are tangent to the parabola defined by $y = x^2$." I believe it's a simple question however I've been going around…
Joel Hernandez
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