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Page 13 Can you see the pattern? And indeed, (Bring the exponent to the front as a coefficient, and reduce the exponent by 1, and use that as the new exponent.) This is our very first formula in derivatives without using the limit definition. It works for any base (as long as the base is the variable and nothing else) and any exponent (as long as the exponent is a real number). Technically, in mathematics, we use the letter n to denote a positive integer, but since this formula applies for any real number exponent, so its better to rewrite the formula as: Use the above formula to find the derivative of the following functions: For negative exponents:For rational exponents:For irrational exponents: Finally a Formula for Derivative and Its Application However, the formula is NOT applicable for the following examples:

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Page 14 Derivative of a Constant Function If f(x) = 7, what is f (x)? And in general, If f(x) = k where k is a constant, what is f (x)? Limits LawsDerivatives Formulas Derivative of a Constant Function and Some Derivative Formulas Why the formula is NOT applicable for the following examples? Some Fundamental Derivative Formulas In limits, we have laws. In derivatives we have formulas too. Applications: 1.If f(x) = 2x 3 – 4x 2 + 5x – 6, what is f (x)?2. Given g(t) = t 6 + t –19 + t –2007, find g (t).3. Keep this in mind too:

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Page 15 Limits LawsDerivatives FormulasIn Words… 1. The limit/derivative of a constant times a function is the constant times the limit/derivative of the function. 2. The limit/derivative of a sum of two functions is the sum of the limit/ derivative of the two functions. 3. The limit/derivative of a difference of two functions is the difference of the limit/derivative of the two functions. Derivative of a Product In the previous page, we showed you three formulas in derivatives, which really are complementary to three laws in limits. In limits, we have the limit of a product of two functions is the product of the limit of the two functions: You might wonder: In derivatives, is the derivative of a product of two functions the product of the derivative of the two functions: If it is, then d / dx [x 2 x 3 ] = But d / dx [(x 2 )(x 3 )] = If it is, then d / dx [2 x 3 ] = But d / dx [2 x 3 ] = f(x)f(x)g(x)g(x) f(x)f(x)g(x)g(x) The two examples above show that the derivative of a product ______ the product of the derivatives. That is,

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Page 16 The Product Rule If the derivative of a product is not the product of the derivatives, then what is the derivative of a product of two functions: It turns out the correct formula should be: and this is called the Product Rule. Keep the second factor function Differentiate the second factor function Differentiate the first factor function Keep the first factor function What did we just do: times plus times Why it works? Lets see, with the product rule, we have: Check! You might say: Why do we need the product rule if its easier without it? Answer: Because we do need it sometimes, maybe a lot of the times, if not always. If you manage to get the derivative of the one above without using the product rule, lets see how you handle this one:

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Page 17 If There Is a Product Rule Then There Is a … If, then what is ? Ans: Definitely NOT. The correct formula is:and this is called the __________________. Lets see how it works on some examples and compare it with the WRONG quotient rule too. The RIGHT Quotient Rulevs.The WRONG Quotient RuleHow Do We Verify Which One Is Right?

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Page 18 Product Rule and Quotient RuleIn Depth Sometimes the function might not be in terms of x, the variable we usually use and differentiate with respect to. For example, the function could be in terms of t, in that case, we would have written the product rule as: Regardless we are using d / dx when we differentiate with respect to x or d / dt when we differentiate with respect to t, we can always use the prime ( ) notation for derivative. Hence, a shorthand for the product rule (by eliminating the xs and ts) is: (f g) = f g + f g and a shorthand for the quotient rule is: Déjà Vu? Example 1: Given f(x) = (2x + 3)(x 2 – 4), find f (x). Example 2: Given f(x) =, find f (x). Different Versions of Product Rule and Quotient Rule: In some textbooks (including ours), the product rule can be written differently as: (f g) = f g + g f(f g) = g f + g f(f g) = g f + f g Its because addition and multiplication are both commutative, i.e., a + b = b + a and ab = ba, hence we have these different versions. For quotient rule, you might see these versions in different textbooks: Our textbook

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Page 19 Product Rule and Quotient RuleTo Use or Not To Use My version of the product rule and quotient rule vs.The textbooks version of the product rule and quotient rule (f g) = f g + f g (f g) = g f + g f If you memorize my version of the product rule, for the quotient rule, all you need to do is to change the + sign the – sign, and put the whole thing over g 2. If you memorize the textbook version of the product rule, it doesnt help you to memorize its version of the quotient rule. That means you need to memorize the quotient rule too, which is obviously more complicated. When to use the product rule and quotient rule and when not to use them? Product RuleExamplesUse it / Dont Use it / Doesnt MatterQuotient RuleExamplesU / DU / DNM 1. f(x) = 2x 3 2. f(x) = ¾(x 2 + 5x – 6) 3. f(x) = (3x – 1)(2x + 4) 4. f(x) = (3x 2 – 2x + 1)(4x 2 + 5x – 4) Change + to –What is the relationship? Dont see any.

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