Skip over navigation

Introduction to Derivatives

Problems for "The Concept of the Derivative"

The Concept of the Derivative

Techniques of Differentiation

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.

Follow Us