Example 3. Find the differential of the function $y=e^{x}\left(x^{2}+3\right)$. Calculate the value of the differential at the point $x=0$.
Solution. We have:

$$
\begin{gathered}
d y=y^{\prime} d x=\left(e^{x}\left(x^{2}+3\right)+2 x e^{x}\right) d x=\left(x^{2}+2 x+3\right) e^{x} d x \\
d y(0)=\left.\left(x^{2}+2 x+3\right) e^{x}\right|_{x=0} d x=3 d x
\end{gathered}
$$

## Exercises

Find the differentials of the functions.

1. $y=\ln \left(\frac{1+x}{1-x}\right)^{1 / 4}-\frac{1}{2} \operatorname{arctg} x$.
2. $y=\frac{3 x^{2}-1}{3 x^{3}}+\ln \sqrt{1+x^{2}}+\operatorname{arctg} x$.
3. $y=\operatorname{arctg} \frac{a}{x}+\ln \sqrt{\frac{x-a}{x+a}}$. 4. $y=\frac{1}{3} \ln \frac{\sqrt{x^{2}+1}-x}{\sqrt{x^{2}+1}+x}$.

Calculate the increments and differentials of the functions
5. $y=2 x^{2}-x$ at $x=1, \Delta x=0.01$.
6. $y=x^{3}+2 x, x=-1, \Delta x=0.02$.

Calculate the approximate values.
7. $\sin 60^{\circ} 18^{\prime}$.
8. $\sqrt[6]{735}$
9. $\operatorname{tg} 46^{\circ}$.
10. $\ln 1.07$

## Answers

1. $d y=\frac{x^{2} d x}{1-x^{4}}$. 2. $d y=\frac{x^{5}+1}{x^{6}+x^{4}} d x$. 3. $d y=\frac{2 a^{3}}{x^{4}-a^{4}} d x$. 4. $d y=-\frac{2 d x}{3 \sqrt{x^{2}+1}}$. 5. $\Delta y=0.0302, d y=0.03$. 6. $\Delta y=0.0988, d y=0.1$. 7. 0.86863 . 8. 3.0041 . 9. 1.03553 . 10. 0.0676586 .

## § 6. Derivatives and Differentials of Higher Orders

$1^{\circ}$. Suppose the derivative $y^{\prime}=f^{\prime}(x)$ of the function $y \doteq f(x)$ is a differentiable function. Then its derivative $\left(y^{\prime}\right)^{\prime}$ is called the second derivative of the function $f(x)$ and is denoted by $y^{\prime \prime}=f^{\prime \prime}(x)$. Higher-order derivatives are defined successively: $y^{\prime \prime \prime}=\left(y^{\prime \prime}\right)^{\prime}, y^{I V}=\left(y^{\prime \prime \prime}\right)^{\prime}$, and so on. Notations can also be: $y^{(1)}, y^{(2)}, y^{(3)}, \ldots, y^{(n)}$

For example, if $y=x^{2}+5 x-6$, then $y^{\prime}=2 x+5, y^{\prime \prime}=2, y^{\prime \prime \prime}=0$.

If $y=\cos x$, then the derivative of any order of this function is found as follows:

$$
\begin{aligned}
& y^{\prime}=(\cos x)^{\prime}=-\sin x=\cos \left(x+\frac{\pi}{2}\right) \\
& y^{\prime \prime}=-\cos x=\cos \left(x+2 \frac{\pi}{2}\right) \\
& y^{\prime \prime \prime}=\sin x=\cos \left(x+3 \frac{\pi}{2}\right) \\
& y^{I V}=\cos x=\cos \left(x+4 \frac{\pi}{2}\right) \\
& \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\
& y^{(n)}=\cos \left(x+n \frac{\pi}{2}\right) ; \quad n=1,2,3, \ldots
\end{aligned}
$$

$2^{\circ}$. Higher-order differentials are defined as follows: $d^{n} y=f^{(n)}(x) d x^{n}$. Hence $f^{(n)}(x)=\frac{d^{n} y}{d x^{n}}=\frac{d}{d x}\left(\frac{d^{n-1} y}{d x^{n-1}}\right)$.

For example, $d^{n}(\cos x)=\cos \left(x+\frac{n \pi}{2}\right) d x^{n}$.

## Exercises

Find the derivatives and differentials of order $n \in \mathbb{N}$.

1. $y=\frac{1}{a x-b}$
2. $e^{3 x}$.
3. $\frac{x^{2}+2 x+3}{x}$
4. $\sqrt{x}$.
5. $x^{\alpha}$.
6. $\ln \sqrt[33]{x+1}$
7. $\sin \alpha x$. 8. $\cos \beta x$.

## Answers

1. $y^{(n)}=(-1)^{n} \frac{a^{n} n!}{(a x-b)^{n+1}}$.
2. $y^{(n)}=3^{n} e^{3 x}$. 3. $y^{\prime}=1-\frac{3}{x^{2}}$, $y^{\prime \prime}=\frac{6}{x^{3}}, \quad y^{(n)}=(-1)^{n} 3 \frac{n!}{x^{n+1}}$
3. $y^{(n)}=(-1)^{n-1} \frac{1 \cdot 3 \cdot 5 \cdots(2 n-3)}{2^{n} \cdot \sqrt{x^{2 n-1}}}$
4. $y^{(n)}=\alpha(\alpha-1) \cdots(\alpha-n+1) x^{\alpha-n}$. 6. $y^{(n)}=\frac{1}{33}(-1)^{n+1} \frac{(n-1)!}{x^{n}}$.
5. $y^{(n)}=\alpha^{n} \sin \left(\alpha x+\frac{\pi}{2} n\right) \cdot 8 \cdot y^{(n)}=\beta^{n} \cos \left(\alpha x+n \frac{\pi}{2}\right)$.

## § 7. Differentiation of Inverse Functions. Differentiation of Functions Given Implicitly and Parametrically

$1^{\circ}$. Let $y=f(x)$ be a function defined and continuous on the interval $[a ; b]$ and $f(a)=A, f