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Compute the derivative of $f(x)=x^n$.


Show that $$\frac{d}{dx} e^x = e^x$$


Given $\frac{d}{dx} e^x = e^x$, find the value of $\frac{d}{dx} \ln x$.


Find the derivative of function $y=\sin{x}$.


Find the derivative of $\arcsin{x}$.


Let $f(x)$ be an odd function which is differentiable over $(-\infty, +\infty)$. Show that $f'(x)$ is even.


Compute the limit of the power series below as a rational function in $x$:

$$1\cdot 2 + (2\cdot 3)x + (3\cdot 4)x^2 + (4\cdot 5)x^3 + (5\cdot 6)x^4+\cdots,\qquad (|x| < 1)$$


Compute $$1-\frac{1\times 2}{2}+\frac{2\times 3}{2^2}-\frac{3\times 4}{2^3}+\frac{4\times 5}{2^4}-\cdots$$


Find the maximum and minimal values of the function

$$f(x)=(x^2-4)^8 -128\sqrt{4-x^2}$$

over its domain.


Find all quadratic polynomials $p(x)=ax^2 + bx + c$ such that graphs of $p(x)$ and $p'(x)$ are tangent to each other at point $(2, 1)$.


Show that $\ln x < \sqrt{x}$ holds for all positive $x$.


Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(x)+f(\frac{1}{x})$.


Show that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ given by

$$f(x,y)=x^4+6x^2y^2 + y^4 -\frac{9}{4}x-\frac{7}{4}$$

achieves its minimal value, and determine all the points $(x, y)\in\mathbb{R}^2$ at which it is achieved.


Note that the point $(2, 1)$ is always on the curve $x^4 + ky^4 = 16+k$ regardless of the value of $k$. If for a particular non-zero value of $k$, $y'(2)=y''(2)$ along this curve. Find this $k$.


Find the derivative of $x^x$.


The equation $x^y=y^x$ describes a curve in the first quadrant of the plane containing the point $P=(4, 2)$. Compute the slope of the line that is tangent to this curve at $P$.


Let $f(x)=x^2\cos(ax)$ where $a$ is a constant. Find the $50^{th}$ order derivative of $f(x)$, i.e. $f^{(50)}(x)$.


Find the number of $k$ such that the function $y=e^{kx}$ satisfies the equation $$\left(\frac{d^2y}{dx^2}+\frac{dy}{dx}\right)\left(\frac{dy}{dx}-y\right)=y\frac{dy}{dx}$$


Find the value of $c$ such that two parabolas $y=x^2+c$ and $y^2=x$ touch at a single point.