# Questions tagged [derivatives]

Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

1,636 questions
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### Discontinuous derivative. [duplicate]

Could someone give an example of a ‘very’ discontinuous derivative? I myself can only come up with examples where the derivative is discontinuous at only one point. I am assuming the function is real-...
1answer
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### How discontinuous can a derivative be?

There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property. To my knowledge, not many "...
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### Why do we require radians in calculus?

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does ...
3answers
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### Second derivative “formula derivation”

I've been trying to understand how the second order derivative "formula" works: $$\lim_{h\to0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}$$ So, the rate of change of the rate of change for an arbitrary ...
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### Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$

Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$. This question is just after the definition of differentiation and the theorem that if $f$ is finitely ...
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### Trapezoid rule error analysis

How can I prove that the max error of the trapezoid rule for the integral $\int_{a}^{b}{f(x)\, \mathrm{d}x}$ is: $$\Delta=-\frac{1}{12n^2}f''(c)(b-a)^3 \text{for } c \in (a,b) \ ?$$ I know that to ...
3answers
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### Failure of differential notation

Through the informal use of differentials, the product rule can be "proved" by writing $$d(fg) = (f + df)(g + dg) - fg = df\,g + f\,dg + df\,dg.$$ Neglecting the product of two differentials, we ...
10answers
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### How is the derivative truly, literally the “best linear approximation” near a point?

I've read many times that the derivative of a function $f(x)$ for a certain $x$ is the best linear approximation of the function for values near $x$. I always thought it was meant in a hand-waving ...
11answers
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### Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?

$1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$. I got this question in an Application of Derivatives test. I think log might be used here to compare the values, but even then the values ...
9answers
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### When can we not treat differentials as fractions? And when is it perfectly OK?

I am a first year calculus student so I would prefer if answers remained in Layman's terms. It is common knowledge and seems to me a mantra that I keep hearing over and over again to "not treat ...
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### What is the formula for nth derivative of arcsin x, arctan x, sec x and tan x?

Are there formulae for the nth derivatives of the following functions? 1) arcsin x 2) arctan x 3) sec x, and 4) tan x Thanks.
2answers
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### Interchanging the order of differentiation and summation

Can the order of a differentiation and summation be interchanged, and if so, what is the basis of the justification for this? e.g. is $\frac{\mathrm{d}}{\mathrm{d}x}\sum_{n=1}^{\infty}f_n(x)$ equal ...
2answers
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### Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
3answers
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### Derivative (or differential) of symmetric square root of a matrix

Let A be a square, symmetric, positive-definite matrix. Let S be its symmetric square root found by a singular value decomposition. Let vech() be the half-vectorization operator. Is there a ...
3answers
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### Derivative of an even function is odd and vice versa

This is the question: "Show that the derivative of an even function is odd and that the derivative of an odd function is even. (Write the equation that says f is even, and differentiate both sides, ...
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### A convex function is differentiable at all but countably many points

Let $f:\Bbb R\to\Bbb R$ be a convex function. Then $f$ is differentiable at all but countably many points. It is clear that a convex function can be non-differentiable at countably many points, for ...
6answers
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### Finding the Derivative of |x| using the Limit Definition

Please Help me derive the derivative of the absolute value of x using the following limit definition. $$\lim_{\Delta x\rightarrow 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ I have no idea as to how to ...
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### Derivative of Quadratic Form

For the Quadratic Form $X^TAX; X\in\mathbb{R}^n, A\in\mathbb{R}^{n \times n}$ (which simplifies to $\Sigma_{i=0}^n\Sigma_{j=0}^nA_{ij}x_ix_j$), I tried to take the derivative wrt. X ($\Delta_X X^TAX$) ...
2answers
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### Proving that if $|f''(x)| \le A$ then $|f'(x)| \le A/2$

Suppose that $f(x)$ is differentiable on $[0,1]$ and $f(0) = f(1) = 0$. It is also known that $|f''(x)| \le A$ for every $x \in (0,1)$. Prove that $|f'(x)| \le A/2$ for every $x \in [0,1]$. I'll ...
3answers
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### A continuously differentiable map is locally Lipschitz

Let $f:\mathbb R^d \to \mathbb R^m$ be a map of class $C^1$. That is, $f$ is continuous and its derivative exists and is also continuous. Why is $f$ locally Lipschitz? Remark Such $f$ will not be ...
2answers
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### Continuous function with linear directional derivatives=>Total differentiability?

As in the title: If $f\colon\mathbb{R}^n\to\mathbb{R}$ is continuous in $x$ and has directional derivatives $\partial_vf(x)=L(v)\,\forall v\in\mathbb{R}^n$, where $L$ is linear, does this imply that \$...