3.2 Derivatives of Polynomials, Exponential Functions, Trigonometric Function, The product rule

1-4. Find the derivative where is













5. Find where .



15/14*x^(1/14) + 5*e^x

 6. Find the equation of the tangent line to the curve at .


  .  So the slope of the tangent line is 20.

       (Since passes through .





 7. The normal line to a curve at a point is the line that passes through and is perpendicular to the tangent line to at . Find an equation of the normal line to the curve at  the point .



      So the slope of the tangent line is . (Since )

      Then the slope of the normal line is .

      Thus, .  

      Therefore, the normal line is .





8. Where is the function differ-entiable? Give a formula for .


      then .

      So is continuous on .

      , then is not differentiable at .

      Therefore, is differentiable on .

9. Let . Find the values of and that make differentiable everywhere.

Note that is differentiable every where except .

      To be differentiable at , , so .

      And also  have to be continuous at .

      , since , . Therefore,


10. Evaluate .


 Method (1) Let , and . Then by the definition of derivative,


      Method (2) Note that .








11-12. Differentiate the following functions.









13. If satisfies the identity for all and and , then show that satisfies for all .


      Thus, or .

      Since for all and .



      Since , then .

      Therefore, .

14-16. Find the following derivatives.

14. .



15. .


16. .


17. Show that the curve has no tangent line with slope .


      Since is always positive, there is no such that .

      So has no tangent line with slope 0.

18. A stone is thrown into a pond, creating wave whose radius increases at the rate of meter per second. In square meter per second, how fast is the area of the circular ripple increasing seconds after the stone hits the water?

 radius time





      Therefore, .

19. A particle moves along a straight line with equation of motion .

    (a) When is the particle moving forward?

    (b) When is (the acceleration)   zero?

    (c) When is the particle speeding up? Slowing down?

    (a) .

    (b) , .

        Therefore, .

    (c) At time ,  (the acceleration) is zero.

       In , the particle is speeding up, in , the particle is slowing down.

20. A particle moves in a straight line with equation of motion , where is measured in second and in meters.

    (a) What is the position of the particle at and ?

    (b) Find the velocity of the particle at time .

    (c) When is the particle moving forward ?

    (d) Find the total distance traveled by particle on the time interval .

    (e) Find the acceleration of the particle at time .

    (a) .

    (b) .

    (c) When , .

       Therefore, .



    (e) .

21. The population of the bacteria colony after hours is . Find the growth rate when .




22. A cost function is given by


    (a) Find the marginal cost function.

    (b) Find .

    (a) .

    (b) .

23. If a stone is thrown vertically upward with a velocity , then its height after seconds is .

    (a) What is the maximum height reached by the stone?

    (b) What is the velocity of the stone when it is above the ground on its way up? On its way down?








       The time is when the height of the stone is 5.


24. If is the total value of the production when there are workers in a plant, then the average productivity is


    Find . Explain why the company wants to hire more worker if ?


      If , since , then .

       is the rate of productivity.


      This means rate of productivity is larger than average productivity .

      Thus, the company wants to hire more workers.

25. Let be the population of bacteria colony at time hours. Find the growth rate of the bacteria after 10 hours.


26. The angular displacement of simple pendulum is given by with the angular amplitude , the angular frequency and a phase constant . Find .


27. Show that .

 Let . Then , and as , .