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Our expert and friendly tutors can help you learn Calculus with ease! Please call us tollfree at 1.877.TUTOR18 or register online to get help in Calculus.

  •   Limit of a function
  •   Continuity and discontinuity of functions
  •   Differentiation
  •   Differentiation of trigonometric functions
  •   Differentiation of implicit functions
  •   Differentiation of parametric functions  
  •   Differentiation of exponential functions
  •   Derivatives of second order  
  •   Monotonic functions  

Our proven 4-step learning approach enables students to understand mathematical concepts and apply these concepts to successfully solve practical problems. We equip you with tips for tricky questions, and teach shortcuts to improve testing speed and mental calculations.

Our examples below illustrate our comprehensive explanations with all steps. This means: Better understanding and Greater success in Calculus for YOU

1.
Differentiate

      Then,

                       


2.
If
, then prove that .

By using chain rule, we get
                 
                 

                 



3. Find the gradient of the curves at their points of intersection. Hence find the angles at which the curves cut.

To find the points of intersection we can solve the given equations simultaneously.

We have been given that
                                    , substituting this in,

We have,
                                         
When

Now we need to fin the gradients, given

                                                     

 

 

Therefore,
                                         
Therefore,

At (0, 0), and at (4, 2),

            Again,

 

            Therefore,
                                           
      Therefore,
                         At (0, 0),

It is clear that at (0, 0) the curves cut at right angles as the tangents to the curves at (0, 0) are at right angles to each other, being parallel and perpendicular to the x-axis respectively.
    
     At the point (4, 2) the angle of intersection is given by the equation