Example 6. Calculate approximately $1.02^{3.01}$.
Solution. Consider the function $z=x^{y}$. The desired number can be considered as the incremental value of this function at $x=1, y=3$, $\Delta x=0.02, \Delta y=0.01$.

The initial value of the function $z=1^{3}=1$. Based on formulas (7.4) and (7.5), we get

$$
\Delta z \approx d z=y x^{y-1} \Delta x+x^{y} \ln x \Delta y=3 \cdot 1 \cdot 0.02+1 \cdot \ln 1 \cdot 0.01=0.06
$$

By formula (7.6), we find

$$
1.02^{3.01} \approx 1+0.06=1.06
$$

## Problems

Find the total differentials of the functions:

1. $z=\sin ^{2} y+\cos ^{2} x$.
2. $z=\frac{x+y}{x y}$.
3. $r=e^{\frac{i}{s}}$.
4. $u=r^{2} \cos ^{2} \varphi$.
5. $u=x^{2} y z$.
6. $u=\ln \left(x^{2}+y^{2}+z^{2}\right)$.

Calculate the values of the total differentials of the functions:
7. $z=\frac{y}{x}$ at $x=2, y=1, d x=0.1, d y=0.2$.
8. $z=e^{x y}$ at $x=1, y=2, d x=-0.1, d y=0.1$.

Approximately calculate the changes in the functions:
9. $u=\operatorname{arctg} \frac{y}{x}, x$ changes from 2 to 2.1; $y$ from 3 to 2.5.
10. $\varphi=\arcsin \frac{y}{x}, x$ changes from 5 to 4.5; $y$ from 3 to 3.3.

Approximately calculate:
11. $(1.02)^{3}(0.97)^{2}$.

$$
\text { 12. } \sqrt{(4.05)^{2}+(2.93)^{2}} \text {. }
$$

13. How will the diagonal $l$ of a rectangle with sides $a=10 \text{cm}, b=24 \text{cm}$ change if side $a$ is extended by 4 mm and side $b$ is shortened by 1 mm? Find the approximate value of the change and compare it with the exact value.

Answers

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-248.jpg?height=111&width=1015&top_left_y=487&top_left_x=87)
$\left.-\frac{t}{s^{2}} d s\right)$
4. $d u=2 r(\cos 2 \varphi d r-r \sin 2 \varphi d \varphi)$.
5. $d u=x(2 z y d x+x z d y+$ $+x y d z)$ 6. $d u=\frac{2}{x^{2}+y^{2}+z^{2}}(x d x+y d y+z d z)$. 7. $0.075 . \quad$ 8. $-0.1 e^{2} \approx$ $\approx-0.739$. 9. $-0.1$. $\quad 10.0.15$.
11. 1.00 .
12. 4.998 .
13. $d l=0.062 \text{cm} ;$ $\Delta l=0.065 \text{cm}$.

## § 7.2. Higher-Order Derivatives and Differentials

The second-order partial derivatives of the function $z=f(x, y)$ are the partial derivatives of its first-order partial derivatives. Notations:

$$
\left.\begin{array}{r}
\frac{\partial^{2} z}{\partial x^{2}}=\frac{\partial}{\partial x}\left(\frac{\partial z}{\partial x}\right) ; \frac{\partial^{2} z}{\partial y^{2}}=\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial y}\right) ; \\
\frac{\partial^{2} z}{\partial x \partial y}=\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right) ; \frac{\partial^{2} z}{\partial y \partial x}=\frac{\partial}{\partial x}\left(\frac{\partial z}{\partial y}\right)
\end{array}\right\}
$$

Other notations are also used:

$$
f_{x x}^{\prime \prime}=\frac{\partial}{\partial x}\left(\frac{\partial z}{\partial x}\right) ; f_{y y}^{\prime \prime}=\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial y}\right), f_{x y}^{\prime \prime}=\frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)
$$

If the mixed derivatives $f_{x y}^{\prime \prime}$ and $f_{y x}^{\prime \prime}$ are continuous, the results of differentiation do not depend on the order of differentiation, i.e.,

$$
\frac{\partial^{2} z}{\partial x \partial y}=\frac{\partial^{2} z}{\partial y \partial x}
$$

Similarly, higher-order partial derivatives are defined. For example, for the function $z=f(x, y)$, the third-order partial derivatives are defined by the formulas:

$$
\begin{gathered}
\frac{\partial^{3} z}{\partial x^{3}}=\frac{\partial}{\partial x}\left(\frac{\partial^{2} z}{\partial x^{2}}\right) ; \frac{\partial^{3} z}{\partial y^{3}}=\frac{\partial}{\partial y}\left(\frac{\partial^{2} z}{\partial y^{2}}\right) ; \\
\frac{\partial^{3} z}{\partial x^{2} \partial y}=\frac{\partial}{\partial y}\left(\frac{\partial^{2} z}{\partial x^{2}}\right) ; \frac{\partial^{3} z}{\partial y^{2} \partial x}=\frac{\partial}{\partial x}\left(\frac{\partial^{2} z}{\partial y^{2}}\right) \text { and so on. }
\end{gathered}
$$

Third-order and higher-order partial derivatives that differ only in the order of differentiation are equal if they are continuous. For example,

$$
\frac{\partial^{3} z}{\partial x^{2} \partial y}=\frac{\partial^{3} z}{\partial y \partial x^{2}}=\frac{\partial^{3} z}{\partial x \partial y \partial x}
$$

The second, third, and higher-order differentials of the function $z=f(x, y)$ are defined by the formulas $d^{2} z=d(d z), d^{3} z=$ $=d\left(d^{2} z\right)$, and so on. They are expressed through partial derivatives as follows:

$$
\begin{gathered}
d^{2} z=\frac{\partial^{2} z}{\partial x^{2}} d x^{2}+2 \frac{\partial^{2} z}{\partial x \partial y} d x d y+\frac{\partial^{2} z}{\partial y^{2}} d y^{2} \\
d^{3} z=\frac{\partial^{3} z}{\partial x^{3}} d x^{3}+3 \frac{\partial^{3} z}{\partial x^{2} \partial y} d x^{2} d y+3 \frac{\partial^{3} z}{\partial x \partial y^{2}} d x d y^{2}+\frac{\partial^{3} z}{\partial y^{3}} d y^{3}
\end{gathered}
$$

In general, the symbolic formula is valid

$$
d^{n} z=\left(d x \frac{\partial}{\partial x}+d y \frac{\partial}{\partial y}\right)^{n} z
$$

which is formally expanded according to the binomial law.