Derivative Formulas

1. If $c$ is a constant, then $D_x(c)=0$. Moreover, $D_x(cy)=cD_x(y)$.

2. 
   
   $$D_x(y+z) = D_x(y)+D_x(z).$$
   
   

3. For any real $n$,
   
   $$D_x(y^n) = ny^{n-1}D_x(y).$$
   
   
   As a special case:
   
   $$D_x(x^n) = nx^{n-1}.$$
   
   

4. 
   
   $$D_x(yz) = D_x(y)z + yD_x(z).$$
   
   

5. 
   
   $$D_x(f(g(x))=f'(g(x))g'(x).$$
   
   

6. Linear approximation formula: Linear approximation to $f(x)$ at $x=a$ is:

   
   

   $$L(x)=f(a)+f'(x)(x-a).$$
   
   

7. Newton's Method for finding a root. Change the current guess $x_n$ to
   
   $$x_{n+1}=x_n - \frac{f(x_n)}{f'(x_n)}.$$