Example 8. Find the derivative of the function $y=\frac{\cos ^{2} x}{\sin x}$.
Solution. Differentiating it as a quotient and a composite function, we find

$$
\begin{aligned}
y^{\prime}= & \left(\frac{\cos ^{2} x}{\sin x}\right)^{\prime}=\frac{\left(\cos ^{2} x\right)^{\prime} \sin x-\cos ^{2} x(\sin x)^{\prime}}{\sin ^{2} x}= \\
& =\frac{2 \cos x(-\sin x) \sin x-\cos ^{2} x \cos x}{\sin ^{2} x}= \\
& =-\frac{\cos x\left(\sin ^{2} x+\cos ^{2} x\right)+\cos x \sin ^{2} x}{\sin ^{2} x}= \\
= & -\cos x\left(\frac{1}{\sin ^{2} x}+1\right)=-\cos x\left(\operatorname{cosec}^{2} x+1\right)
\end{aligned}
$$

## Problems

Find the derivatives of the functions:

1. $y=\sin \frac{x}{2}$.
2. $y=\cos (3-4 x)$.
3. $y=\cos ^{3} x$.
4. $y=\sin ^{4} x+\cos ^{4} x$.
5. $y=\sqrt{x^{4}+2 x+3}$.
6. $y=\left(2 x^{2}-7\right)^{3}$.
7. $y=\left(a^{2}-c x^{2}\right)^{3}$.
8. $y=\frac{1}{(a x+b)^{2}}$.
9. $y=\frac{1}{(1+\sin 2 x)^{3}}$.
10. $y=-\operatorname{ctg}^{3} x+3 \operatorname{ctg} x+3 x$.
11. $y=\frac{\sin ^{2} x}{\cos x}$.
12. $y=\cos ^{2} x^{2}$.

## Answers

1. $\frac{1}{2} \cos \frac{x}{2}$.
2. $4 \sin (3-4 x)$.
3. $-3 \cos ^{2} x \sin x$.
4. $-\sin 4 x$.
5. $\frac{2 x^{3}+1}{\sqrt{x^{4}+2 x+3}}$.
6. $12 x\left(2 x^{2}-7\right)^{2}$. 7. $-6 c x\left(a^{2}-c x^{2}\right)^{2}$.
7. $-\frac{2 a}{(a x+b)^{3}}$.
8. $\frac{-6 \cos 2 x}{(1+\sin 2 x)^{4}}$. 10. $3 \operatorname{ctg}^{4} x \cdot$ 11. $\sin x\left(\sec ^{2} x+1\right)$. 12. $-2 x \sin 2 x^{2}$.

## § 2.3. Derivatives of Exponential and Logarithmic Functions

Basic formulas:

$$
\begin{gathered}
\left(a^{x}\right)^{\prime}=a^{x} \ln a \\
\left(e^{x}\right)^{\prime}=e^{x} \\
\left(\log _{a} x\right)^{\prime}=\frac{1}{x} \log _{a} e=\frac{1}{x \ln a} \\
(\ln x)^{\prime}=\frac{1}{x}
\end{gathered}
$$

If $z=z(x)$ is a differentiable function of $x$, then the formulas become:

$$
\begin{gathered}
\left(a^{z}\right)^{\prime}=a^{z} \cdot \ln a \cdot z^{\prime} \\
\left(e^{z}\right)^{\prime}=e^{z} \cdot z^{\prime} \\
\left(\log _{a} z\right)^{\prime}=\frac{z^{\prime}}{z \ln a} \\
(\ln z)^{\prime}=\frac{z^{\prime}}{z}
\end{gathered}
$$