Derivative values at given points and solving f'(x) = 0

Let

find $f'(1)$, $f'(2)$ and $f'(3)$.

Let

find $f'(2)$.

Let

find $f'(1)$.

Using the table of derivatives, find the derivative of the following function:$y=2+x-x^{2}$. What are $y'(0)$; $y'\left(\dfrac {1}{2}\right)$; $y'(1)$; $y'(-10)$?

Using the table of derivatives, find the derivative of the following function:$y=\dfrac {x^{3}}{3}+\dfrac {x^{2}}{2}-2x$. For what value of $x$ is:

$y=f(x^{2})$;

Find $f^{\prime }\left(x_{0}\right)$ if:

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = (x+1)(x+2)(x+3)(x+4), x_{0} = -3$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = (x-a)(x-b)(x-c), x_{0} = a$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = \frac{x-a}{x-b}, a \neq b, x_{0} = a$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = \left(1+a x^{b}\right)\left(1+b x^{a}\right), x_{0} = 1$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = x(x-1)(x-2) \ldots (x-1984)(x-1985), x_{0} = 0, x_{0} = 1985$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = \left(2-x^{2}\right) \cos x+2 x \sin x, x_{0} = 0$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = \frac{\sin x-x \cos x}{\cos x+x \sin x}, x_{0} = \frac{\pi }{2}$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = (a x+b) \cos x+(c x+d) \sin x, x_{0} = 0$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = \operatorname {arctg} x \cdot \arccos x, x_{0} = 0$

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = \log_{2} x \ln 2 x, x_{0} = 1$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = \frac{x^{2}}{\ln x}, x_{0} = e$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = x^{5} e^{-x}, x_{0} = 5$.

Compute the derivative of the function $y = f(x)$ at the point $x_{0}$. $y = x \operatorname {sh} 2 x, x_{0} = 1$.

Compute the derivative of the function at the indicated points. $y = \frac{1+x-x^{2}}{1-x+x^{2}}, x = 0, x = 1$.

Compute the derivative of the function at the indicated points. $y = (1+x) \sqrt{2+x^{2}} \sqrt[3]{3+x^{3}}, x = 0$.

Compute the derivative of the function at the indicated points. $y = 2^{\operatorname {tg}(1 / x)}, x = 1 / \pi$.

Compute the derivative of the function at the indicated points. $y = 3 \cos 2 x-\sqrt{1-\sin 2 x}(\sin x+\cos x), x = \pi / 6$.

Compute the derivative of the function at the indicated points. $y = \log_{1 / 2}(x-1 / 2)^{2}+\log_{2} \sqrt{4 x^{2}-4 x+1}, x = 0$.

Compute the derivative of the function at the indicated points. $y = \sqrt{\ln x}\left(\ln x-\log_{e x} x\right) \sqrt{\ln x+\log_{x} e+2}, x = e$.

Compute the derivative of the function at the indicated points. $y = \ln \left(1+\sin^{2} x\right)-2 \sin x \cdot \operatorname {arctg} \sin x, x = \pi / 2$.

Compute the derivative of the function at the indicated points. $y = \arcsin \left(2 x /\left(1+x^{2}\right)\right), x = 0, x = 2$.

Compute the derivative of the function at the indicated points. $y = \arcsin \left(\left(1-x^{2}\right) /\left(1+x^{2}\right)\right), x = -1, x = 1, x = 0$.

Compute the derivative of the function at the indicated points. $y = \ln \frac{x^{4}-x^{2}+1}{x^{4}+2 x^{2}+1}+2 \sqrt{3} \operatorname {arctg} \frac{\sqrt{3}}{1-2 x^{2}}, x = 1$.

Compute the derivative of the function at the indicated points. $y = \sqrt[3]{\operatorname {arctg} \sqrt[5]{\cos \ln^{3} x}}, x = 1$.

Solve the equation $y^{\prime }(x) = 0$ if: