Higher-Order Derivatives and Differentials

$y = x^{2}+13 x+11$;

$y = 1+10 x+\frac{1}{x^{98}}$;

$y = \frac{x\left(1+3 \sqrt{1-x^{2}}\right)}{\sqrt{1-x^{2}}}$;

$y = \cos^{2} x$;

$y = \operatorname {sh}^{2} x+\operatorname {ch}^{2} x$;

$y = \ln \left(x+\sqrt{x^{2}+1}\right)$;

$y = \operatorname {arctg}\left(x+\sqrt{x^{2}+1}\right)$;

$y = \operatorname {arcctg} \frac{1-x}{\sqrt{2 x-x^{2}}}$;

$y = \arcsin \frac{x^{2}-1}{x^{2}+1}$;

$y = 2 \operatorname {arctg} \frac{\sqrt[4]{x^{4}+1}}{x}-\ln \frac{\sqrt[4]{1+x^{4}}+x}{\sqrt[4]{1+x^{4}}-x}$.

$y = e^{\sqrt{x}}, x = 4$;

$y = \frac{x^{5}}{(x-1)^{4}}, x = 5$;

$y = \frac{\arcsin x}{\sqrt{1-x^{2}}}, x = 0$;

$y = e^{\sin x} \cos (\sin x), x = 0$.

$y = \left(x^{2}+x+1\right) e^{-x}$;

$y = 2 x+\operatorname {ctg} 2 x$;

$y = x(\cos \ln x+\sin \ln x)$;

$y = x^{x}$.

$y = \frac{x^{3}}{3-x^{2}}, x = 1$;

$y = \frac{3 x+2}{x^{2}-2 x+5}, x = 0$;

$y = x \sqrt[3]{(x-5)^{2}}, x = -3$;

$y = \operatorname {arctg} \frac{2+x^{2}}{2-x^{2}}, x = 0$.

$y = A \cos a x+B \sin a x, y^{\prime \prime }+a^{2} y = 0$

$y = A e^{a x}+B e^{-a x}, y^{\prime \prime }-a^{2} y = 0$

$y = (A \cos 3 x+B \sin 3 x) e^{-x}, y^{\prime \prime }+2 y^{\prime }+10 y = 0$;

$y = A e^{x}+B e^{-x}-\frac{1}{x}, y^{\prime \prime }-y = \frac{x^{2}-2}{x^{3}}$

$y = 1+\cos e^{x}+\sin e^{x}, y^{\prime \prime }-y^{\prime }+e^{2 x} y = 0$;

$y = \left(x+\sqrt{1+x^{2}}\right)^{10},\left(1+x^{2}\right) y^{\prime \prime }+x y^{\prime }-100 y = 0$;

$y = e^{10 \arcsin x},\left(1-x^{2}\right) y^{\prime \prime }-x y^{\prime }-100 y = 0$;

$y = \cos (10 \arccos x),\left(1-x^{2}\right) y^{\prime \prime }-x y^{\prime }+100 y = 0$.

$y = (v+2 u) / u$;

$y = e^{u v}$;

$y = \operatorname {arctg}(v / u)$;

$y = \ln \sqrt{u^{2}+v^{2}}$.

$y = u(2+v)$

$y = u \ln v$

$y = \sqrt{u^{2}+v^{2}}$;

$y = u^{v}$.

$x = t^{3}, y = t^{2}$;

$x = \frac{t^{2}}{1+t^{3}}, y = \frac{t^{3}}{1+t^{3}}$;

$x = \ln \cos t, y = \ln \cos 2 t$;

$x = a \cos t, y = b \sin t$;

$x = \left(1+\cos^{2} t\right) \sin t, y = \sin^{2} t \cos t$

$x = t \operatorname {ch} t-\operatorname {sh} t, y = t \operatorname {sh} t-\operatorname {ch} t$;

$x = \frac{e^{t}}{1+t}, y = (t-1) e^{t}$;

$x = \frac{1}{\cos t}, y = \operatorname {tg} t-t$;

$x = \sin \log_{2} t, y = \operatorname {tg} \log_{2} t$;

$x = 2^{\cos^{2} t}, y = 2^{\sin^{2} t}$.

$x = \left(t^{2}+1\right) e^{t}, y = t^{2} \mathrm{e}^{2 t},(1 ; 0)$;

$x = \frac{2 t-t^{2}}{t-1}, y = \frac{t^{2}}{t-1},(0 ; 4)$;

$x = \ln (1+\sin \varphi ), y = \ln (1-\cos 2 \varphi ),(\ln (3 / 2) ; \ln (1 / 2))$;

$x = \operatorname {ch} t \sin t+\operatorname {sh} t \cos t, y = \operatorname {ch} t \cos t-\operatorname {sh} t \sin t,(0 ; 1)$.

$x = -2+3 t-t^{3}, y = t+2 t^{2}+t^{3}$;

$x = \ln \operatorname {tg}(t / 2), y = \ln \operatorname {tg} t$;

$x = \log_{5} \sin t, y = \log_{5} \cos t$;

$x = \arcsin \operatorname {tg} t, y = \sqrt{\cos 2 t}$.

$x = t^{3}+t, y = \frac{3}{4} t^{4}+\frac{1}{2} t^{2}+1, y^{\prime \prime }\left(1+3 y^{2}\right) = 1$;

$x = e^{t} \cos t, y = e^{t} \sin t,-\pi / 4 < t < \pi / 4, \quad (x-y)^{2} y^{\prime \prime } = 2\left(x y^{\prime }-y\right)$;

$x = \sin t, y = A e^{\sqrt{2} t}+B e^{-\sqrt{2} t},-\pi / 2 < t < \pi / 2$, $A$ and $B$ are arbitrary constants, $\left(1-x^{2}\right) y^{\prime \prime }-x y^{\prime }-2 y = 0$;

$x = 3 A t^{2}+\ln B t, y = 2 A t^{3}+t, A$ and $B$ are arbitrary positive constants, $\left(3 y-2 y^{\prime }\right) y^{\prime \prime } = y^{\prime 2}$. -ations:

$x = a \cos t, y = a \sin t$;

$x = a \operatorname {ch} t, y = a \operatorname {sh} t$

$x = a \cos^{3} t, y = a \sin^{3} t$;

$x = a(t-\sin t), y = a(1-\cos t)$;

$x = e^{-t} \cos t, y = e^{-t} \sin t$;

$x = \cos t-\ln \operatorname {ctg}(t / 2), y = \sin t$.

$x = a \cos^{2} t, y = b \sin^{2} t$;

$x = \cos t, y = \cos n t, n \in \mathbb {N}$;

$x = t^{2}-t+1, y = t^{2}+t+1$;

$x = \frac{t}{t+1}, y = \frac{2 t^{2}+t}{(t+1)^{2}}$.

$x^{2}+y^{2} = a^{2}$;

$x^{2}-y^{2} = a^{2}$;

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} = 1$;

$y^{2} = 2 p x$;

$e^{x-y} = x+y$;

$e^{2 y}-2 \ln x-1 = 0$;

$y-x \operatorname {tg} \ln \sqrt{x^{2}+y^{2}} = 0$;

$y^{2} = e^{x^{4}-y^{2}}$.

$x^{2}+2 x y+y^{2}-4 x+2 y-2 = 0,(1 ; 1)$

$2 \ln (y-x)+\sin x y = 0,(0 ; 1)$;

$x^{3} y+\arcsin (y-x) = 1$, $(1 ; 1)$;

$3(y-x+1)+\operatorname {arctg}(y / x) = 0,(1 ; 0)$.

$y = A \ln y+x+B, y \cdot y^{\prime \prime } = \left(y^{\prime }\right)^{2}-\left(y^{\prime }\right)^{3}$;

$(A+B x) e^{y / x} = x, x^{3} y^{\prime \prime } = \left(x y^{\prime }-y\right)^{2}$.

$y = x^{3}+x+e^{3 x}$;

$y = a_{0} x^{n}+a_{1} x^{n-1}+\ldots +a_{n}$;

$y = \frac{1+x}{1-x}$;

$y = \frac{a x+b}{c x+d}$;

$y = \ln (a x+b)$;

$y = \sin^{2} x$;

$y = \sin a x \sin b x$;

$y = \operatorname {ch} a x \operatorname {ch} b x$;

$y = \sin^{2} x \sin 2 x$;

$y = \sin^{4} x+\cos^{4} x$;

$y = \cos^{4} x$

$y = \frac{1}{\sqrt{1-2 x}}$

$y = \frac{x}{x^{2}-4 x-12}$;

$y = \frac{1+x^{2}}{1-x^{2}}$

$y = \frac{3-2 x^{2}}{2 x^{2}+3 x-2}$.

$y = (x-1) 2^{x-1}$;

$y = (2 x-1) 2^{3 x} \cdot 3^{2 x}$;

$y = (3-2 x)^{2} e^{2-3 x}$;

$y = x \log_{2}(1-3 x)$

$y = \ln (x-1)^{2 x}$;

$y = x \ln \frac{3+x}{3-x}$

$y = x \ln \left(x^{2}-3 x+2\right)$

$y = x \cos x$;

$y = 2 x \cos^{2}(x / 3)$;

$y = \left(x^{2}+x\right) \cos^{2} x$.

$y = \frac{x}{\sqrt{1-5 x}}$;

$\frac{x^{2}}{\sqrt{1-2 x}}$;

$y = e^{a x} \cos (b x+c)$;

$y = e^{2 x} \sin^{2} x$

$y = \operatorname {ch} a x \sin b x$

$y = x^{n-1} e^{1 / x}$

$y = x^{n-1} \ln x$

$y = \operatorname {arctg} x$.

$y = (2 x-7)^{2}(3 x+7)^{3}$; a) $n = 5, x = x_{0}$; b) $n = 6, x = x_{0}$;

$y = \sqrt{x} ; n = 10, x = 1$;

$y = \frac{x^{2}}{1-x} ; n = 8, x = 0$;

$y = \frac{x+1}{x^{2}+x-2}$; $n = 13, x = -\frac{1}{2}$;

$y = \sqrt{x^{2}+3 x^{3}} ; n = 5, x = 1$;

$y = \frac{1-x^{2}}{\sqrt[3]{2-3 x}} ; n = 6, x = -1$;

$y = x^{2} \ln x ; n = 100, x = 1$;

$y = \left(x^{2}-2 x\right) \cos 3 x ; n = 101, x = 1$;

$y = x \sin x \cos 2 x ; n = 100, x = \pi / 2$;

$y = (x-\sin x)^{2} ; n = 16, x = \pi / 4$;

$y = \operatorname {arctg}^{2} x$; a) $n = 10, x = 0$; b) $n = 11, x = 0$.

$y = (x+5)^{5} ; n = 3, x = 0$;

$y = \frac{7 x+1}{(3 x-2)^{2}} ; n = 10, x = 1$;

$y = \left(\sqrt{x^{2}-1}+\sqrt{x-1}\right)^{2} ; n = 16, x = 1$;

$y = \sin x \sin 2 x \sin 3 x ; n = 10, x = \pi / 6$;

$y = \left(2 x^{2}+1\right) \operatorname {sh}^{2} x ; n = 8, x = 0$;

$y = \arcsin x$; a) $n = 19, x = 0$; b) $n = 20, x = 0$.

$y(x) = \begin{cases} 1-\cos x, & \text{ if } \quad x < 0 \\ \ln (1+x)-x, & \text{ if } \quad x \geqslant 0 \end{cases}$

$y(x) = \begin{cases} 2 x \cos x, & \text{ if } x < 0, \\ \sin 2 x, & \text{ if } x \geqslant 0 ;\end{cases}$

$y(x) = \begin{cases} \operatorname {sh} x-x, & \text{ if } x < 0 \\ x-\sin x, & \text{ if } x \geqslant 0\end{cases}$

$y(x) = \begin{cases} \operatorname {sh} x, & \text{ if } x < 0 \\ \sin x \operatorname {ch} x, & \text{ if } x \geqslant 0 \end{cases}$

$y(x) = \begin{cases} x^{10}, & \text{ if } x \text{ — rational number, } \\ -x^{10}, & \text{ if } x \text{ — irrational number; } \end{cases}$

$y(x) = \begin{cases} x^{100} \sin (1 / x), & \text{ if } x \neq 0 \\ 0, & \text{ if } x = 0 \end{cases}$

$y(x) = \begin{cases} e^{-1 / x^{2}}, & \text{ if } x \neq 0, \\ 0, & \text{ if } x = 0. \end{cases}$