7.2.4 Exercises

Find the following derivatives by simplifying the terms of the functions and then applying your knowledge of the differentiation of common functions ($x>0$):

1. $f(x):=x^6·x^{\frac{7}{2}}=$ .
   
2. $g(x):=\frac{x^{-\frac{3}{2}}}{\sqrt{x}}=$ .
   

Thus, we have:

1. $f\text{'}(x)=$ .
   
2. $g\text{'}(x)=$ .
   

Solution

Simplify the terms of the functions and find their derivatives:

1. $f(x):=2\sin \left(\frac{x}{2}\right)·\cos \left(\frac{x}{2}\right)=$ .
   
2. $g(x):={\cos }^2(3x)+{\sin }^2(3x)=$ .
   

Thus, we have:

1. $f\text{'}(x)=$ .
   
2. $g\text{'}(x)=$ .
   

Solution

Simplify the terms of the functions and find the derivatives (for $x>0$ in the first part of this exercise):

1. $f(x):=3\ln (x)+\ln \left(\frac{1}{x}\right)=$ .
   
2. $g(x):={\left(e{}^x\right)}^2·e{}^{-x}=$ .
   

Thus, we have:

1. $f\text{'}(x)=$ .
   
2. $g\text{'}(x)=$ .
   

Enter the exponential functions as exp, for example, enter $e{}^{4x^2}$ as exp(4*x^2).
Solution