Derivative - Hyperbolic Functions

Category: Differential Calculus, Algebra

"Published in Newark, California, USA"

Find the derivative for

Solution:

Consider the given equation above

Take the derivative of the above equation with respect to x, we have

We can accept the above equation as a final answer but if you wish to substitute the value of hyperbolic cosine, you can also do that one. We know that 

If 

then

becomes

The other way of getting the derivative of the given equation is to eliminate the hyperbolic functions by substituting their equivalent value or identity. We know that 

If

then substitute the value of hyperbolic sine of the above equation as follows

Take the derivative of the above equation with respect to x, we have

Triple Integration

Category: Integral Calculus, Trigonometry

"Published in Newark, California, USA"

Evaluate

Solution:

Consider the given equation above

Integrate first the given function with respect to dz, as follows

Integrate the above equation with respect to dr, as follows

Integrate the above equation with respect to dθ, as follows

Substitute the limits and the final answer is 

  

Double Integration

Category: Integral Calculus

"Published in Newark, California, USA"

Evaluate 

Solution:

Consider the given equation above

Integrate first the given function with respect to dy, as follows

If u = y, then du = dy. The above equation can be integrated into inverse trigonometric function

Integrate the above equation with respect to dx, as follows

Substitute the limits and the final answer is