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When finding the first derivative, you calculate the slope of the curve at a point by determining the ratio of delta y over delta x. You make delta x and delta y smaller and smaller; in fact, you take the limit of delta y over delta x as delta x goes to zero.
At this zero limit point, delta y is called dy, and delta x is called dx. They are also called differentials, hence the term differential calculus. They represent the rate at which x and y change at various x's and y's.
Since we are working with differentials, the process of determining the slope, also known as the first derivative, is call differentiation.
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The process of finding the product of two numbers?
It is called multiplication.
What is the process for finding the product of two integer?
It is called multiplication.
What is the derivative of y3?
I understand this to be y3. Then the derivative is 3y2. If y is considered a so-called 'implicit function' of x then the derivative might be written 3y2 dy/dx.
What is the second derivative of x raised to the 1 divided by x?
To calculate the derivate of a power, where both the base and the exponent are functions of x, requires a technique called logarithmic derivation. I'll leave the details to you; it is not particularly difficult. You can look up "logarithmic differentiation" in the Wikipedia for some examples.
