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Saturday, August 3, 2013

Derivative - Trigonometric Functions, 3

Category: Differential Calculus, Trigonometry

"Published in Suisun City, California, USA"

Prove that


Solution:

Consider the above equation


Rewrite the left side of the equation as a quotient of two trigonometric functions as follows


Take the derivative of the above equation using the quotient of the two functions formula, we have





But


Hence, the above equation becomes




Therefore,

Friday, August 2, 2013

Derivative - Trigonometric Functions, 2

Category: Differential Calculus, Algebra, Trigonometry

"Published in Newark, California, USA"

Find the second derivative for


Solution:

Consider the given equation above


Since the denominator of the given equation contains a radical sign, then we have to rationalize the denominator in order to eliminate the radical sign as follows





But



Hence, the above equation becomes





Using the Half Angle Formula, the above equation becomes




Take the derivative on both sides of the equation with respect to x, we have





Take the derivative on both sides of the equation again with respect to x, we have







Therefore, the final answer is


Note: You must memorize or remember the trigonometric formulas and identities as much as you can so that it will be easier for you to take the derivative of trigonometric functions and equations.

Thursday, August 1, 2013

Integration - Powers

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate


Solution:

Consider the given equation above


The terms inside the radical sign has a common factor, which is x. Factor the terms inside the radical sign, we have



The other factor is a perfect trinomial square. Rewrite the above equation in terms of a square of a binomial



Take the square root of the above equation


Apply the distributive property of multiplication over addition to the above equation


Integrate the above equation by power with respect to x, we have









Therefore,


where C is the constant of integration.