Chapter 7 Differential Calculus

Section 7.3 Calculation Rules

7.3.5 Exercises

Exercise 7.3.9
Calculate the derivatives of the functions $f$, $g$, and $h$ defined by the following mapping rules:
1. The derivative of $f\left(x\right):=3+5x$ is $f\text{'}\left(x\right)=$ .
2. The derivative of $g\left(x\right):=\frac{1}{4x}-{x}^{3}$ is $g\text{'}\left(x\right)=$ .
3. The derivative of $h\left(x\right):=2\sqrt{x}+4{x}^{-3}$ is $h\text{'}\left(x\right)=$ .

Exercise 7.3.10
Calculate the derivatives of the functions $f$, $g$, and $h$ described by the following mapping rules, and simplify the results.
1. The derivative of $f\left(x\right):=\mathrm{cot}x=\frac{\mathrm{cos}x}{\mathrm{sin}x}$ is $f\text{'}\left(x\right)=$ .
2. The derivative of $g\left(x\right):=\mathrm{sin}\left(3x\right)·\mathrm{cos}\left(3x\right)$ is $g\text{'}\left(x\right)=$ .
3. The derivative of $h\left(x\right):=\frac{\mathrm{sin}\left(3x\right)}{\mathrm{sin}\left(6x\right)}$ is $h\text{'}\left(x\right)=$ .

Exercise 7.3.11
Calculate the derivatives of the functions $f$, $g$, and $h$ defined by the following mapping rules:
1. The derivative of $f\left(x\right):=e{}^{5x}$ is $f\text{'}\left(x\right)=$ .
2. The derivative of $g\left(x\right):=x·e{}^{6x}$ is $g\text{'}\left(x\right)=$ .
3. The derivative of $h\left(x\right):=\left({x}^{2}-x\right)·e{}^{-2x}$ is $h\text{'}\left(x\right)=$ .

Exercise 7.3.12
Calculate the first four derivatives of $f:ℝ\to ℝ$ with $f\left(x\right):=\mathrm{sin}\left(1-2x\right)$.
Answer: The $k$th derivative of $f$ is denoted by ${f}^{\left(k\right)}$. Here, ${f}^{\left(1\right)}=f\text{'}$, ${f}^{\left(2\right)}$ is the derivative of ${f}^{\left(1\right)}$, ${f}^{\left(3\right)}$ is the derivative of ${f}^{\left(2\right)}$, etc. Thus, we have:
• ${f}^{\left(1\right)}\left(x\right)=$ .
• ${f}^{\left(2\right)}\left(x\right)=$ .
• ${f}^{\left(3\right)}\left(x\right)=$ .
• ${f}^{\left(4\right)}\left(x\right)=$ .