The derivative of the function \( f\left(x_{i}\right)=x^{3} \) is equal to \( f^{\prime}(x)=3 x^{2} \)

4. Let \( f(x)=\ln \left(x^{2}+1\right) \), then \( f^{\prime}(x)= \) Let \( f(x)=e^{x} \), then \( f^{n}(x)= \) The function \( f(x)=-2 x^{3}+a x^{2}+b x+c \) the turnig point of \( p(8,-9) \) and \( (5,18) \), then the value of \( a= \) and \( b= \).

Let \( f(x)=x^{2}+15 \), then \( f^{10}(x)= \) Let \( f(x)=\ln \left(x^{2}+1\right) \), then \( f^{\prime}(x)= \) Let \( f(x)=e^{\infty x} \), then \( f^{n}(x)= \)

A. The critical number of \( f(x) \) B. Find the intervals on which \( f \) is increasing and decreasing by using the first derivative test Find the local maximum and minimum, if they exist by using the first derivative test Find the intervals on which \( f \) is concave up and concave down Find the inflection points of \( f(x) \)

What is the solution to the following differential equation, subject to the initial condition \( y(0)=2 \) ? \( \frac{d y}{d x}=6 x+6 \)

20. \( \int x \tan ^{2} x d x \)