Derivative Functions

(1) Proposition (Derivatives of the Trigonometric Functions) The trigonometric functions sine, cosine, tangent, cotangent, cosecant, and secant are all differentiable functions on their domain and their derivative functions are:

        

        

        

    Proof. For the derivative of the cosine function, we use the formula



along with the definition of the derivative:















For the derivative of the sine function, we use the formula



along with the definition of the derivative:















For the derivative of the tangent function, we use the formula along with the quotient rule:



For the derivative of the cotangent function, we use the formula   along with the quotient rule:



For the derivative of the secant function, we use the formula   along with the quotient rule:



For the derivative of the cosecant function, we use the formula   along with the quotient rule:

    Since the trigonometric functions are differentiable functions on their domains they are also continuous functions on their domain.

(2) Example (Derivatives of the Trigonometric Functions) Find the derivative functions for the functions and

    Solution. For the function we use the quotient rule, derivative rules for sine and cosine, and a few trigonometric identities, we determine,
    




and simplifies to,   For the function we use the quotient rule and the derivative rules for sine and cosine, we determine,

(3) Proposition (Derivatives of the Inverse Trigonometric Functions) The inverse trigonometric functions arcsine, arccosine, arctangent, arccotangent, arccosecant, and arcsecant are all differentiable functions on their domain and their derivative functions are:

        

        

    

(4) Example (Derivatives of the Inverse Trigonometric Functions) Find the derivative of the function

    Solution. Using the product rule and the derivative formulas for arcsine and arccosine we determine:





(5) Example (Derivatives of the Inverse Trigonometric Functions) Find the derivative of the function

    Solution. Using the product rule and the derivative formulas for arctangent and arccotangent we determine:
    
         



(6) Example (Derivatives of the Inverse Trigonometric Functions) Find the derivative of the function

    Solution. Using the product rule and the derivative formulas for arctangent and arcsecant we determine:
    



(7) Example (Derivatives of the Inverse Trigonometric Functions) Find the derivative of the function ;      

    Solution. Using the quotient rule, the derivative formulas for arcsine and arccosine and some trigonometric identities, we determine:     







Recall, the cofunction theorem from trigonometry: if and then if and only if

(8) Proposition (Derivatives of Exponential Functions) The derivative of the exponential function is In the special case when we have and So,

        

(9) Example (Derivatives of Exponential Functions) Find the equation of the tangent line to the function at

    Solution. Using the product rule, the derivative of   is



and so



The equation of the tangent line has slope



and so we have and with the point and thus,  



Therefore, an equation of the tangent line is



Here is an illustration of the graph of and the tangent line:

(10) Proposition (Derivatives of Logarithmic Functions) The derivative of the logarithmic function is In the special case when we have and So,

        

(11) Example (Derivatives of Logarithmic Functions) Find the equation of the tangent line to the curve at

    Solution. The derivative of is and at we have  the slope of the tangent line as Therefore, the equation of the tangent line is which simplifies to
    



(12) Example (Derivatives of Logarithmic Functions) For what values of and does satisfy

    Solution. We determine,
    




Since we find that and that can be any real number.

(13) Definition (Rectilinear Motion) An object that moves along a straight line with position has velocity and acceleration when these derivatives exist. The speed of an object at time is

(14) Example (Rectilinear Motion) A particle moving along the -axis has position



after an elapsed time of seconds.

(a) Find the velocity of the particle at time

    Solution. The velocity is given by



(b) Find the acceleration at time

    Solution. The acceleration is given by
    


(c) What is the total distance travelled by the particle during the first 3 seconds?

    Solution. Since when



but is not on the distance covered is

.

(15) Proposition (Falling Body Problem) The position of a free-falling body (neglect air resistance) under the influence of gravity can be represented by the function



where is the acceleration due to gravity (on earth ) and and are the initial height and velocity of the object (when ).

(16) Example (Falling Object Problem) A ball is thrown vertically upward from the ground with an initial velocity of 160 ft/s.

(a) When will it hit the ground?

    Solution. Since with and we need



This is precisely when or and thus the ball will hit the ground in 10 seconds.  

(b) With what velocity will the ball hit the ground?

    Solution. The velocity function is and so ft/sec is the velocity when the ball will hit the ground.

(c) When will the ball reach its maximum height?

    Solution. Since when or when The ball will reach its maximum height at seconds.

• print this page
• pages linking here
• related pages

Calculus for Dummies List Price: $19.99 Buy New: $9.55 You Save: $10.44 (52%) New (58) Used (54) from $7.50 The mere thought of having to take a required calculus course is enough to make legions of students break out in a cold sweat. Others who have no intention of ever studying the subject have this notion (more)

Cracking the AP Calculus AB & BC Exams, 2008 Edition (College Test Prep) List Price: $19.00 Buy New: $12.05 You Save: $6.95 (37%) New (26) Used (12) from $10.86 Scoring high on the AP Calculus AB & BC Exams is very different from earning straight A?s in school. We don?t try to teach you everything there is to know about calculus?only the strategies and information (more)

Barron's AP Calculus (Barron's How to Prepare for Ap Calculus Advanced Placem... List Price: $16.99 Buy New: $7.80 You Save: $9.19 (54%) New (22) Used (18) from $6.11 Both Calculus AB and Calculus BC are covered in this comprehensive AP test preparation manual. Prospective test takers will find four practice exams in Calculus AB and four more in Calculus BC, with all (more)

Student Solutions Manual for Single Variable Calculus: Early Transcendentals ... List Price: $61.95 Buy New: $44.04 You Save: $17.91 (29%) New (18) Used (9) from $41.00 Student Solutions Manual for Single Variable Calculus: Early Transcendentals and Calculus: Early Transcendental (more)

Calculus Workbook For Dummies (Dummies Series) List Price: $16.99 Buy New: $5.45 You Save: $11.54 (68%) New (43) Used (23) from $5.00 From differentiation to integration - solve problems with ease Got a grasp on the terms and concepts you need to know, but get lost halfway through a problem or, worse yet, not know where to begin? (more)