Example. Find the derivative of the function $y=x^{e^{x}} x^{9}$.

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Translation:
Example. Find the derivative of the function $y=x^{e^{x}} x^{9}$.
Solution.

1. The logarithm of the given function is

$$
\ln y=\ln \left(x^{e^{x}} x^{9}\right)=e^{x} \ln x+9 \ln x
$$

2. Differentiating both sides of this equation, we get

$$
\frac{y^{\prime}}{y}=e^{x} \ln x+e^{x} \frac{1}{x}+\frac{9}{x}
$$

Therefore,

$$
y^{\prime}=y\left(e^{x} \ln x+\frac{e^{x}+9}{x}\right)
$$

3. Substituting the expression for $y$ into the last equation, we get the answer.

Answer. $y^{\prime}=x^{e^{x}} x^{9}\left(e^{x} \cdot \ln x+\frac{e^{x}+9}{x}\right)$.

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

1. $y=(\sin x)^{\sqrt{x}}$.
2. $y=x^{x} 3^{\sqrt{x}}$.
3. $y=x^{e^{\sqrt{x}}}$.
4. $y=(\operatorname{arctg} x)^{x}$.
5. $y=(\operatorname{tg} x)^{\ln x}$.
6. $y=x^{\sqrt{x}} 2^{\sin x}$.
7. $y=x^{2^{\cos x}}$.
8. $y=x^{\operatorname{tg} x}$.
9. $\quad y=(\sin 3 x)^{\ln \sin 3 x}$.
10. $y=x^{2^{x}} 3^{x}$.

## Answers.

1. $y^{\prime}=y\left(\frac{\ln \sin x}{2 \sqrt{x}}+\frac{\sqrt{x} \cos x}{\sin x}\right)$.
2. $y^{\prime}=y\left(\ln x+1+\frac{\ln 3}{2 \sqrt{x}}\right)$.
3. $y^{\prime}=y e^{\sqrt{x}}\left(\frac{\ln x}{2 \sqrt{x}}+\frac{1}{x}\right)$.
4. $y^{\prime}=y\left[\ln \operatorname{arctg} x+\frac{x}{\left(1+x^{2}\right) \operatorname{arctg} x}\right]$.
5. $y^{\prime}=y\left(\frac{\ln \operatorname{tg} x}{x}+\frac{\ln x}{\sin x \cos x}\right)$.
6. $y^{\prime}=y\left(\frac{\ln x}{2 \sqrt{x}}+\frac{1}{\sqrt{x}}+\cos x \ln 2\right)$.
7. $y^{\prime}=y 2^{\cos x}\left(-\sin x \ln 2+\frac{1}{x}\right)$.
8. $y^{\prime}=y\left(\frac{\ln x}{\cos ^{2} x}+\frac{\operatorname{tg} x}{x}\right)$.

![](https://cdn.mathpix.com/cropped/2024_05_22_5878ce417e2ca061000bg-107.jpg?height=91&width=1045&top_left_y=954&top_left_x=119)

## 4.6. Derivative of a function given parametrically

Problem Statement. Find the derivative of the function given parametrically.

Solution Plan. If the dependence of $y$ on $x$ is given through a parameter $t$:

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

then the dependence of $y^{\prime}$ on $x$ is given through the parameter $t$ by the formulas

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

We compute $f^{\prime}(t)$ and $g^{\prime}(t)$, substitute into formula (1), and write the answer.