7.3.5 Exercises

Calculate the derivatives of the functions $f$, $g$, and $h$ defined by the following mapping rules:

1. The derivative of $f(x):=3+5x$ is $f\text{'}(x)=$ .
   
2. The derivative of $g(x):=\frac{1}{4x}-x^3$ is $g\text{'}(x)=$ .
   
3. The derivative of $h(x):=2\sqrt{x}+4x^{-3}$ is $h\text{'}(x)=$ .
   

Solution

Calculate the derivatives of the functions $f$, $g$, and $h$ described by the following mapping rules, and simplify the results.

1. The derivative of $f(x):=\cot x=\frac{\cos x}{\sin x}$ is $f\text{'}(x)=$ .
   
2. The derivative of $g(x):=\sin (3x)·\cos (3x)$ is $g\text{'}(x)=$ .
   
3. The derivative of $h(x):=\frac{\sin (3x)}{\sin (6x)}$ is $h\text{'}(x)=$ .
   

Solution

Calculate the derivatives of the functions $f$, $g$, and $h$ defined by the following mapping rules:

1. The derivative of $f(x):=e{}^{5x}$ is $f\text{'}(x)=$ .
   
2. The derivative of $g(x):=x·e{}^{6x}$ is $g\text{'}(x)=$ .
   
3. The derivative of $h(x):=(x^2-x)·e{}^{-2x}$ is $h\text{'}(x)=$ .
   

Solution

Calculate the first four derivatives of $f:ℝ\to ℝ$ with $f(x):=\sin (1-2x)$.
Answer: The $k$th derivative of $f$ is denoted by $f^{(k)}$. Here, $f^{(1)}=f\text{'}$, $f^{(2)}$ is the derivative of $f^{(1)}$, $f^{(3)}$ is the derivative of $f^{(2)}$, etc. Thus, we have:

• $f^{(1)}(x)=$ .
  
• $f^{(2)}(x)=$ .
  
• $f^{(3)}(x)=$ .
  
• $f^{(4)}(x)=$ .
  

Solution