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,

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.

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.