Example. Find the fourth-order derivative of the function

$$
y=\left(x^{3}+2\right) e^{4 x+3}
$$
## Solution.

1. We apply Leibniz's formula (1). In this case,

$$
n=4, \quad u(x)=x^{3}+2, \quad v(x)=e^{4 x+3}
$$

We have

$$
\begin{gathered}
u^{\prime}(x)=3 x^{2}, \quad u^{\prime \prime}(x)=6 x, \quad u^{\prime \prime \prime}(x)=6, \quad u^{(4)}(x)=0 \\
v^{\prime}(x)=4 e^{4 x+3}, \quad v^{\prime \prime}(x)=4^{2} e^{4 x+3}, \quad v^{\prime \prime \prime}(x)=4^{3} e^{4 x+3}, \quad v^{(4)}(x)=4^{4} e^{4 x+3}
\end{gathered}
$$

Substituting the obtained results into formula (1), we get

$$
\begin{aligned}
& y^{\mathrm{iv}}=\left(\left(x^{3}+2\right) e^{4 x+3}\right)^{(4)}=0 \cdot e^{4 x+3}+4 \cdot 6 \cdot 4 \cdot e^{4 x+3}+ \\
&+\frac{4 \cdot 3}{1 \cdot 2}(6 x) 4^{2} e^{4 x+3}+\frac{4 \cdot 3 \cdot 2}{1 \cdot 2 \cdot 3}\left(3 x^{2}\right) 4^{3} e^{4 x+3}+\frac{4 \cdot 3 \cdot 2 \cdot 1}{1 \cdot 2 \cdot 3 \cdot 4}\left(x^{3}+2\right) 4^{4} e^{4 x+3}= \\
&= 32 e^{4 x+3}\left(8 x^{3}+24 x^{2}+18 x+19\right)
\end{aligned}
$$

Answer. $y^{(4)}=32 e^{4 x+3}\left(8 x^{3}+24 x^{2}+18 x+19\right)$.

Conditions of the Problems. Find the derivatives of the specified order of the given functions.

1. $y=\left(x^{2}+1\right) \ln x, y^{(5)}=$ ?
2. $y=\left(x^{3}+2\right) \cos 2 x, \quad y^{\prime \prime \prime}=$ ?
3. $y=x^{2} \sin (3 x+1), y^{\prime \prime \prime}=$ ?
4. $y=\left(x^{2}+1\right) \cos 3 x, \quad y^{(5)}=$ ?
5. $y=\left(x^{3}+x\right) \ln x, \quad y^{\prime \prime \prime}=$ ?
6. $y=\left(x^{3}+1\right) \ln (x+2), \quad y^{(5)}=$ ?
7. $y=\left(x^{2}+1\right) 2^{x}, y^{(5)}=$ ?
8. $y=\sin 2 x \ln x, \quad y^{\prime \prime \prime}=$ ?
9. $y=\left(x^{2}+1\right) e^{2 x+1}, y^{(5)}=$ ?
10. $y=\left(x^{3}+2\right) 3^{3 x}, \quad y^{(4)}=?$

## Answers.

1. $y^{(5)}=\frac{x^{2}+24}{x^{5}}$.
2. $y^{\prime \prime \prime}=\left(8 x^{3}-36 x+16\right) \sin 2 x-\left(36 x^{2}-6\right) \cos 2 x$.
3. $y^{\prime \prime \prime}=\left(18-27 x^{2}\right) \cos (3 x+1)-54 x \sin (3 x+1)$.
4. $y^{(5)}=810 x \cos 3 x-27\left(9 x^{2}-11\right) \sin 3 x$.
5. $y^{\prime \prime \prime}=\frac{6 x^{2} \ln x+11 x^{2}-1}{x^{2}}$.
6. $y^{(5)}=-\frac{6 x^{3}+60 x^{2}+240 x+456}{(x+2)^{5}}$.
7. $y^{(5)}=2^{x} \ln ^{3} 2\left[\left(x^{2}+1\right) \ln ^{2} 2+10 x \ln x+20\right]$.
8. $y^{\prime \prime \prime}=-8 \cos 2 x \ln x-\frac{6 \cos 2 x}{x^{2}}-\frac{\left(12 x^{2}-2\right) \sin 2 x}{x^{3}}$.
9. $y^{(5)}=32 e^{2 x+1}\left(x^{2}+5 x+6\right)$.
10. $y^{(4)}=3^{3 x+2} \ln 3\left[\left(9 x^{3}+18\right) \ln ^{3} 3+36 x^{2} \ln ^{2} 3+36 x \ln 3+8\right]$.

### 4.10. Second Derivative of a Function Given Parametrically

Problem Statement. Find the second-order derivative of a function given parametrically.

Solution Plan. If the function is given parametrically:

$$
\left\{\begin{array}{l}
x=f(t) \\
y=g(t)
\end{array}\right.
$$

then its first derivative is determined by the formulas

$$
\left\{\begin{array}{l}
x=f(t) \\
y^{\prime}=\frac{g^{\prime}(t)}{f^{\prime}(t)}
\end{array}\right.
$$

Differentiating $y^{\prime}$ with respect to $x$ as a composite function of $x$ and using the formula for the derivative of the inverse function, we get

$$
\left\{\begin{array}{l}
x=f(t) \\
y^{\prime \prime}=\left(\frac{g^{\prime}(t)}{f^{\prime}(t)}\right)^{\prime} \cdot \frac{1}{f^{\prime}(t)}
\end{array}\right.
$$