Assume that \(\displaystyle f:(a,b)\to \R\) is differentiable, \(\displaystyle \lim_{x\to b}f(x)=\infty\). Does it imply that \(\displaystyle \lim_{x\to b}\) \(\displaystyle f'(x)\) \(\displaystyle =\) \(\displaystyle \infty\)?

Difficulty: 2.

\(\displaystyle \Bigg(\sin\bigg({\sin x\over \sqrt{x}}\bigg)\Bigg)'=?\)

Difficulty: 2.

Let \(\displaystyle f:\R\to \R\) be differentiable, \(\displaystyle \lim_{x\to \infty} f=1\). Does it imply that \(\displaystyle \lim_{x\to \infty} f'=0\)? And if we also know that \(\displaystyle \lim_{x\to \infty}f'\) exists?

Difficulty: 2.

Calculate the derivative:

\(\displaystyle -x; \qquad 3x^3-2x+1; \qquad \frac{x^2+1}{x^3+2}; \qquad (x^{10}+x^2+1)^{100}; \qquad \dfrac{(x^3+1)^n}{(2+x)\bigg(x^3+\dfrac2{x^2}\bigg)} \)

Difficulty: 2.

Calculate the derivative:

\(\displaystyle \dfrac{(x^2+1)^4(2-x)^8}{x^3+2}\cdot\dfrac{1+\dfrac1{1+x}}{2-x} \)

Difficulty: 2.

The following functions are derivatives. For which functions?

\(\displaystyle 1+x+x^2; \qquad x+\frac1x; \qquad \frac{x^2}{(x^3+1)^2} \)

Difficulty: 2.

Calculate the derivative of both sides of the identity

\(\displaystyle 1+x+x^2+\ldots+x^n = \dfrac{1-x^{n+1}}{1-x} \qquad (x\ne1). \)

Difficulty: 2.

\(\displaystyle a)\ \ (x^x)'=?\qquad\qquad b)\ \ \big((\sin x)^{\cos x}\big)'=?\)

Difficulty: 3.

Where is the function

\(\displaystyle f(x)=\begin{cases} \ \ x^2 & \text{if $x\in \Q$} \\ -x^2 & \text{if $x\not\in \Q$} \end{cases}\)

differentiable?

Difficulty: 3.

Where is the function \(\displaystyle \left( \{ x\} -\frac12 \right) ^2 \) differentiable?

Difficulty: 3.

Where is the function \(\displaystyle f(x)=\frac{x}{|x|+1}\) differentiable? What is the derivative?

Difficulty: 3.

Let \(\displaystyle f(x)=x^2 ,\) if \(\displaystyle x\leq 1,\) és \(\displaystyle f(x)=ax+b,\) ha \(\displaystyle x>1.\) For which values of \(\displaystyle a\) and \(\displaystyle b\) will\(\displaystyle f\) be differentiable?

Difficulty: 3.

Prove that the function \(\displaystyle f(x)=\sqrt x\) is differentiable for all \(\displaystyle a>0\) and \(\displaystyle f'(a)=1/(2\sqrt a ).\)

Difficulty: 3.

Assume that \(\displaystyle f:\R\to\R\) is differentiable everywhere. Prove that if \(\displaystyle f\) is even, then \(\displaystyle f'\) is odd and vice versa.

Difficulty: 3.

Calculate the derivative:

\(\displaystyle \sin x^2 \qquad e^{\tg x} \qquad \log_3(\ctg^2x) \qquad \arc\tg(x^2+1) \qquad \sin\Big(\arch\big(\arc\cos(\log_5x)\big)\Big) \)

Difficulty: 3.

\(\displaystyle 8x+\cos x\) is strictly monotone increasing. What is the derivative of its inverse in \(\displaystyle 1\)?

Difficulty: 3.

\(\displaystyle x^x\) is strictly monotone increasing in \(\displaystyle [1,\infty )\). What is the derivative of its inverse in \(\displaystyle 27\)?

Difficulty: 3.

\(\displaystyle x^5 +x^2\) is strictly monotone increasing in \(\displaystyle [1,\infty )\). What is the derivative of its inverse in \(\displaystyle 2\)?

Difficulty: 3.

Prove that \(\displaystyle x+\sin x\) is strictly monotone increasing in \(\displaystyle [1,\infty )\). What is the derivative of its inverse in \(\displaystyle 1+(\pi /2)\)?

Difficulty: 3.

Calculate the derivative of the following functions.

\(\displaystyle x^3; \qquad 2^x; \qquad \log_{1/2}x; \qquad \frac1{\sqrt{x}}; \qquad e^x+3\ln x \qquad x^23^x \)

\(\displaystyle \frac{\sin x}x \qquad x^3e^x\cos x; \qquad x^3\cdot\left(\frac12\right)^x; \qquad \frac{x^2\cdot\ln x\cdot 3^x\cdot\cos x}{\sqrt x -\frac{3\sin x}{x^3}}. \)

Difficulty: 3.

What is the derivative of the inverse function of \(\displaystyle x^5+x^3\) at the point \(\displaystyle -2\)?

Difficulty: 3.

Prove that if \(\displaystyle f(a)=g(a)\) and \(\displaystyle f(x)\leq g(x)\) in a neighborhood of \(\displaystyle a\), then \(\displaystyle f'(a)=g'(a)\).

Difficulty: 3.

Calculate the derivative of the following functions.

\(\displaystyle x^2e^{x^2+\cos x^2} \qquad \log_{\cth^2x+1}\ctg\frac{5^{\tg x}}{\ch x} \qquad \frac{\displaystyle\frac{2^{\ln x/2}}x+\arcth x}{\root3\of{x}+\root5\of{x}} \qquad \frac{\displaystyle \frac{\tg x}{x^2+1}\cdot \frac{\sqrt{x}\cdot 10^x}{\log_3x+x\ctg x} }{\displaystyle (x+1)(x^2+x^e)\cos x} \)

Difficulty: 3.

\(\displaystyle \big(f(x)^{g(x)}\big)'=? \qquad \big(\log_{f(x)}g(x)\big)'=? \)

Difficulty: 3.

Assume that \(\displaystyle f:(a,b)\to \R\) is differentiable and \(\displaystyle \lim_{b} f(x)=\infty\). Does it imply that \(\displaystyle \lim_{b} f'(x) = \infty\)?

Difficulty: 3.

Calculate the derivative!

 (1)  \(\displaystyle \sin\big({\sin x\over \sqrt{x}}\big)\),      (2)  \(\displaystyle x^x\),      (3)  \(\displaystyle (\sin x)^{\cos x}\).

Difficulty: 3.

Let \(\displaystyle f(x)=x\cdot (x+1)\cdots (x+100),\) and let \(\displaystyle g=f\circ f\circ f.\) Calculate \(\displaystyle g'(0)\).

Difficulty: 4.

Let \(\displaystyle f(x)=x^2 \cdot \sin (1/x),\ f(0)=0.\) Prove that \(\displaystyle f\) is differentiable everywhere.

Difficulty: 4.

Prove that \(\displaystyle x^x\) is differentiable for all \(\displaystyle x>0\). Calculate the derivative!

Difficulty: 4.

Find a function \(\displaystyle f(x)\) for which \(\displaystyle f'(0)=0\), and not differentiable at any other points.

Difficulty: 4.

Prove that if \(\displaystyle f'(x)=x^2\) for all \(\displaystyle x\) then there is a constant \(\displaystyle c\) such that \(\displaystyle f(x) =(x^3 /3) +c\).

Difficulty: 4.

Prove that if \(\displaystyle f(a)=g(a)\) and for \(\displaystyle x>a\) we have \(\displaystyle f'(x)\geq g'(x),\) then \(\displaystyle f(x)\geq g(x)\) for all \(\displaystyle x>a\).

Difficulty: 4.

Find a function \(\displaystyle f\) such that \(\displaystyle \lim\limits_{x\to\infty}f'(x)=0,\) but \(\displaystyle \lim\limits_{x\to\infty}f(x)\neq 0.\)

Difficulty: 4.

Assume that      (1)  \(\displaystyle x\cdot f(x), \ \ \ \ \)  (2)  \(\displaystyle f(x^3), \ \ \ \ \)  (3)  \(\displaystyle f^3(x)\)

is differentiable at 0. Does it imply that \(\displaystyle f(x)\) is differentiable at 0?

Difficulty: 4.

Let

\(\displaystyle f(x)=\begin{cases} x+2x^2 \cdot \sin\frac1x , & \text{if $x \ne 0 ,$} \cr 0, & \text{if $x = 0.$}\cr \end{cases} \)

Show that \(\displaystyle f'(0)>1,\) but \(\displaystyle f\) is not monotone increasing in any neighborhood of \(\displaystyle 0\).

Difficulty: 4.

Suppose that \(\displaystyle f\) is differentiable and \(\displaystyle |f'|<K\). Then \(\displaystyle f\) is uniformly continuous.

Difficulty: 4.

Prove that the graph of the function

\(\displaystyle f(x)= \begin{cases} x^x & \text{if $x>0$} \\ 0 & \text{if $x=0$} \end{cases} \)

is tangent to the \(\displaystyle y\)-axis.

Difficulty: 4.

Prove that if \(\displaystyle f'(x)=f(x)\) for all \(\displaystyle x\) then there is a constant \(\displaystyle c\), such that \(\displaystyle f(x)=c\cdot e^x\).

Difficulty: 5.

Calculate the derivative of the Chebishev polynomials at \(\displaystyle 1\): \(\displaystyle T_n'(1)=?\)  \(\displaystyle U_n'(1)=?\)

Difficulty: 5.

Is there a function \(\displaystyle f:\R\to\R\) such that \(\displaystyle f'(x)=\infty\) for all \(\displaystyle x\)?

Difficulty: 5.

Is it true that if \(\displaystyle f\) is continuous in \(\displaystyle a\) and \(\displaystyle \lim_{x\to a} f'(x)=\infty\), then \(\displaystyle f'(a)=\infty\)"?

Difficulty: 5.

\(\displaystyle \sum_{n=1}^{\infty} {n^3\over 3^n}=?\)

Difficulty: 5.

Prove that if \(\displaystyle f\) is differentiable at \(\displaystyle a\), then

\(\displaystyle \lim_{h\to 0}\frac{f(a+h)-f(a-h)}{2h}=f'(a). \)

Show that the statement cannot be reversed.

Difficulty: 5.

Prove that if \(\displaystyle f'(x)\geq \frac1{100}\), then \(\displaystyle \lim\limits_{x\to\infty}f(x)=\infty\).

Difficulty: 6.

Find an everywhere differentiable function with a non continuous derivative!

Check the Darboux theorem for the derivative!

Difficulty: 6.

Let \(\displaystyle [a,a+\delta)\subset D(f)\). Put the following quantities in increasing order:

\(\displaystyle \overline{f_+'}(a) \qquad \underline{f_+'}(a) \qquad \Olim_{a+0}\overline{f'} \qquad \Olim_{a+0}\underline{f'} \qquad \Ulim_{a+0}\overline{f'} \qquad \Ulim_{a+0}\underline{f'} \)

Difficulty: 7.

Does there exists a monotone \(\displaystyle \R\to\R\) function which is not differentiable at any point?

Difficulty: 10.