Problem : f (x) = 2x. Find f'(x)


f'(x) =  
  =  
  =  
  = 2  
  = 2  

Problem : f (x) = - x2. What is the f'(x)?


f'(x) =  
  =  
  =  
  = - (2x + h)  
  = - 2x  

Problem : Find the slope of the line perpendicular to the tangent of the graph of f (x) = - x2 at x = 1.


f'(1) = - 2.
This is the slope of the tangent. The perpendicular to the tangent has a slope that is the negative reciprocal of this, which is .

Problem : Find f'(1) for f (x) = - x2 without solving for f'(x) first.


If we are trying to find the derivative at a single point only, we can substitute that point directly into the limit of the difference quotient


f'(1) =  
  =  
  =  
  =  
  = - (2 + h)  
  = - 2  

Problem : Give an example of a function that is not differentiable at x = a.


f (x) =|x - a| is not differentiable at x = a because it has a corner there.