The product rule states that if h(x) = f(x)g(x) then h'(x) = f'(x)g(x) + f(x)g'(x). In other words if a function is made by multiplying two functions, say the first function and the second function, the derivative of the new function equals the derivative of the first function times the second function + the derivative of the second function time the first function. For example the derivative of (3x^2)(5x^3) is 6x(5x^3) + (3x^2)(15x^2). From ch 4.1 we know tha the derivative of 3x^2 = 6x. 1. How could you use the power rule to take the derivative of 3x^2 ? 2. Do you get the same answer? 3. Which is the easiest way to do this derivative, the power rule or the product rule ?
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
The product rule states that if h(x) = f(x)g(x) then h'(x) = f'(x)g(x) + f(x)g'(x). In other words if a function is made by multiplying two functions, say the first function and the second function, the derivative of the new function equals the derivative of the first function times the second function + the derivative of the second function time the first function.
For example the derivative of (3x^2)(5x^3) is 6x(5x^3) + (3x^2)(15x^2).
From ch 4.1 we know tha the derivative of 3x^2 = 6x.
1. How could you use the power rule to take the derivative of 3x^2 ?
2. Do you get the same answer?
3. Which is the easiest way to do this derivative, the power rule or the product rule ?
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