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Wednesday, March 5, 2014

Integration - Algebraic Functions, Powers, 15

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate


Solution:

Consider the given equation above


The first thing that we have to do is to square the binomial, we have





















Therefore,



where C is the constant of integration.

Tuesday, March 4, 2014

Integration - Algebraic Functions, Powers, 14

Category: Integral Calculus, Algebra

"Published in Newark, California, USA"

Evaluate


Solution:

Consider the given equation above


The first thing that we need to do is to rewrite the radical sign into its equivalent fractional exponent as follows










Therefore,


where C is the constant of integration.

Monday, March 3, 2014

Second Derivative Problems - Chain Rule

Category: Differential Calculus, Algebra

"Published in Newark, California, USA"

If y = f(u) and u = ϕ(x), show that 


Solution:

Consider the given two equations above



Since y is a function of u and u is a function of x, then we can apply the derivative of y with respect to x by Chain Rule Method.

Take the derivative of the first equation with respect to u, we have



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



Hence, by Chain Rule Method,



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










but


Hence, the above equation becomes





Therefore,


where