Derivative - Algebraic, Rational Function

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Find the derivative for

Solution:

Consider the given equation above

Take the derivative of the given equation with respect to y, we have

Apply the derivative by power at the right side of the equation

Next, apply the derivative of rational function at the right side of the equation

Coterminal Angles

Category: Trigonometry

"Published in Suisun City, California, USA"

Find an angle between 0° and 360° that is coterminal with the given angle:

a. 2223°
b. -400°
c. 1270°
d. -800°

Solution:

Coterminal angles are angles in standard position (angles with the initial side on the positive x-axis) that have a common terminal side (angles less than 360°). Usually, coterminal angles are more than 360°. To simplify a coterminal angle, divide a given angle by 360° which is equivalent to 1 revolution. The remainder or a fraction of a revolution will be the terminal side. Let's have the examples of the following:

a. 2223°

    Since the given angle is positive, then it is rotated counterclockwise from positive x-axis (initial side) to terminal side.

After six complete revolution from positive x-axis in a counterclockwise direction, there's an excess angle of 

which is located at the first quadrant.

Photo by Math Principles in Everyday Life

b. - 400°

    Since the given angle is negative, then it is rotated clockwise from positive x-axis (initial side) to terminal side.

After one complete revolution from positive x-axis in a clockwise direction, there's an excess angle of

which is located at the fourth quadrant.

Photo by Math Principles in Everyday Life

c. 1270°

    Since the given angle is positive, then it is rotated counterclockwise from positive x-axis (initial side) to terminal side.

After three complete revolution from positive x-axis in a counterclockwise direction, there's an excess angle of 

which is located at the third quadrant.

Photo by Math Principles in Everyday Life

 d. - 800°

 Since the given angle is negative, then it is rotated clockwise from positive x-axis (initial side) to terminal side.

After two complete revolution from positive x-axis in a clockwise direction, there's an excess angle of

 which is located at the fourth quadrant.

 

Photo by Math Principles in Everyday Life

Solving Exponential Equations, 2

Category: Algebra

"Published in Suisun City, California, USA"

Find the value of x for

Solution:

Consider the given equation above

This type of exponential equation is considered a difficult one because their bases are different. You need to take natural logarithm on both sides of the equation in order to solve for the value of x in their exponents. Take natural logarithm on both sides of the equation, we have

Or, if you will use a calculator, x will be equal to