Implicit Differentiation

Category: Differential Calculus, Algebra

"Published in Suisun City, California, USA"

Find y" by implicit differentiation for

Solution:

Consider the given equation above

Since the given equation above is not a function, then we have to differentiate it by implicit differentiation as follows

but

or

then the above equation becomes

Rationalize the denominator in order to eliminate the radical sign at the denominator, we have

Take the derivative with respect to x, we have

Rationalize the denominator in order to eliminate the radical sign at the denominator, we have

Therefore,

Proving Trigonometric Identities, 8

Category: Trigonometry, Algebra

"Published in Suisun City, California, USA"

Prove that

   
Solution: 

Consider the given equation above

In proving the trigonometric functions, the first thing that you have to do is to choose the more complicated side of the equation and then simplify and compare with the other side of the equation if they are equal or not. 

In this case, the left side of the equation is more complicated and we have to simplify it as follows

The numerator is the sum of cubes. We can use the principles of Algebra in factoring the numerator as follows

but

then the above equation becomes

Therefore,

Proving Trigonometric Identities, 7

Category: Trigonometry, Algebra

"Published in Suisun City, California, USA"

Prove that 

Solution:

In proving the trigonometric functions, the first thing that you have to do is to choose the more complicated side of the equation and then simplify and compare with the other side of the equation if they are equal or not. 

In this case, let's choose the left side of the equation as follows

If you will continue to expand further the above equation, it will be more complicated and longer. Let's substitute all  trigonometric functions with another variables as follows

Let 

 

then the above equation becomes

since

then the above equation becomes

Therefore,