6.9. \( f(x)=4 x \ln x+5 \frac{e^{x}}{\cos x} \).
Solution.

It is recommended to use the rules of differentiation for the sum, product, and quotient of functions.

$$
\begin{gathered}
(f(x))^{\prime}=\left(x \ln x-5 \frac{e^{x}}{\cos x}\right)^{\prime}= \\
=\left[(u v)^{\prime}=u^{\prime} v+u v^{\prime},(u / v)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}},(x)^{\prime}=1,\left(e^{x}\right)^{\prime}=e^{x}\right. \\
\left.(\ln x)^{\prime}=1 / x,(\cos x)^{\prime}=-\sin x\right]= \\
=(x)^{\prime} \ln x+x(\ln x)^{\prime}-5 \frac{\left(e^{x}\right)^{\prime} \cos x-e^{x}(\cos x)^{\prime}}{\cos ^{2} x}=1 \cdot \ln x+x \cdot \frac{1}{x}- \\
-5 \cdot \frac{e^{x} \cdot \cos x-e^{x} \cdot(-\sin x)}{\cos ^{2} x}=\ln x+1-5 \cdot \frac{e^{x} \cdot \cos x+e^{x} \cdot \sin x}{\cos ^{2} x}
\end{gathered}
$$

## CALCULATION OF DERIVATIVES OF COMPLEX FUNCTIONS