Examples of product rule in the following topics:
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- The rules of differentiation can simplify derivatives by eliminating the need for complicated limit calculations.
- In many cases, complicated limit calculations by direct application of Newton's difference quotient can be avoided by using differentiation rules.
- Some of the most basic rules are the following.
- Here the second term was computed using the chain rule and the third using the product rule.
- The flight of model rockets can be modeled using the product rule.
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- A useful property of logarithms states that the logarithm of a product of two quantities is the sum of the logarithms of the two factors.
- Logarithms were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables.
- Tedious multi-digit multiplication steps can be replaced by table look-ups and simpler addition, because of the fact that the logarithm of a product is the sum of the logarithms of the factors:
- We can see that this rule is true by writing the logarithms in terms of exponentials.
- Relate the product rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of products
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- We have already seen that the logarithm of a product is the sum of the logarithms of the factors:
- Another way to see that this rule is true is to apply both the power and product rules, and the fact that dividing by $y$ is the same is multiplying by $y^{-1}.$ So we can write $\log_b(x/y)=\log_b(x\cdot y^{-1}) = \log_bx + \log_b(y^{-1}) = \log_bx -\log_by. $
- By applying the product, power, and quotient rules, you could write this expression as $\log_2(x^4)+\log_2(y^9)-\log_2(z^{100}) = 4\log_2x+9\log_2y-100\log_2z.$
- Relate the quotient rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of quotients
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- We have already seen that the logarithm of a product is the sum of the logarithms of the factors:
- If we apply this rule repeatedly we can device another rule for simplifying expressions of the form $\log_b x^p$.
- Then we have $\log_b(x^p) = \log_b (x \cdot x \cdots x) = \log_b x + \log_b x + \cdots +\log_b x = p\log_b x.$ Since the $p$ factors of $x$ are converted to $p $ summands by the product rule formula.
- This can be written as $\log_3 (3^x) + \log_3 9 + \log_3(x^{100}) = x+2+100\log_3 x, $ using a combination of the product and power rules.
- Relate the power rule for logarithms to the rules for operating with exponents, and use this rule to rewrite logarithms of powers
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- The cross product is a binary operation of two three-dimensional vectors.
- The cross product is denoted as $a \times b = c$.
- The direction of vector $c$ can be found by using the right hand rule.
- The cross product is different from the dot product because the answer is in vector form in the same number of dimensions as the original two vectors, where the dot product is given in the form of a single quantity in one dimension.
- If you use the rules shown in the figure, your thumb will be pointing in the direction of vector $c$, the cross product of the vectors.
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- In Marx's view, social stratification is created by people's differing relationship to the means of production: either they own productive property or they labor for others.
- In a capitalist society, the ruling class, or the bourgeoisie, owns the means of production, such as machines or tools that can be used to produce valuable objects.
- The working class, or the proletariat, only possess their own labor power, which they sell to the ruling class in the form of wage labor to survive.
- In a capitalist society, the ruling class promotes its own ideologies and values as the norm for the entire society, and these ideas and values are accepted by the working class.
- The means of production would be shared by all members of society, and social stratification would be abolished.
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- Kirchhoff's loop rule (otherwise known as Kirchhoff's voltage law (KVL), Kirchhoff's mesh rule, Kirchhoff's second law, or Kirchhoff's second rule) is a rule pertaining to circuits, and is based on the principle of conservation of energy.
- Another equivalent statement is that the algebraic sum of the products of resistances of conductors (and currents in them) in a closed loop is equal to the total electromotive force available in that loop.
- Kirchhoff's second rule requires emf−Ir−IR1−IR2=0.
- We justify Kirchhoff's Rules from diarrhea and conservation of energy.
- An example of Kirchhoff's second rule where the sum of the changes in potential around a closed loop must be zero.
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- Kirchhoff's junction rule states that at any circuit junction, the sum of the currents flowing into and out of that junction are equal.
- Kirchhoff's junction rule, also known as Kirchhoff's current law (KCL), Kirchoff's first law, Kirchhoff's point rule, and Kirchhoff's nodal rule, is an application of the principle of conservation of electric charge.
- Thus, Kirchoff's junction rule can be stated mathematically as a sum of currents (I):
- This law is founded on the conservation of charge (measured in coulombs), which is the product of current (amperes) and time (seconds).
- We justify Kirchhoff's Rules from diarrhea and conservation of energy.
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- Previously, we have applied these rules only to expressions involving integers.
- For example, consider the rule for multiplying two numbers with exponents.
- The same rule applies to expressions with variables.
- Applying the rule for dividing exponential expressions with the same base, we have:
- We can also apply the rule for raising a power to another exponent:
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- Formal structure of an organization or group includes a fixed set of rules for intra-organization procedures and structures.
- The formal structure of a group or organization includes a fixed set of rules of procedures and structures, usually set out in writing, with a language of rules that ostensibly leave little discretion for interpretation.
- Formal rules are often adapted to subjective interests giving the practical everyday life of an organization more informality.
- At first this discovery was ignored and dismissed as the product of avoidable errors, until these unwritten laws of were recognized to have more influence on the fate of the enterprise than those conceived on organizational charts of the executive level.
- A formal organization is a fixed set of rules of intra-organization procedures and structures.