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Problems for "The Concept of the Derivative"

Problems for "The Concept of the Derivative"

Problems for "The Concept of the Derivative"

Problems for "The Concept of the Derivative"

Problems for "The Concept of the Derivative"

Problems for "The Concept of the Derivative"

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


f'(x)   =  
    =  
    =  
    = 2  
    = 2  

Problem : f (x) = - x 2 . 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) = - x 2 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) = - x 2 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.