6 ## RULES## FOR## DERIVATIVESThe derivative of a sum or difference The derivative of a constant times a function The derivative of the square root THE DEFINTION of the derivative is fundamental. (Definition 5.) The student should be thoroughly familiar with it. From that definition it is possible to prove various rules, some of which we will present in this Lesson. The student will find it extremely helpful to state each rule verbally. For example,
"The derivative of a constant is 0." We should expect this, because the slope of a horizontal line
"The derivative of a variable with respect to itself is 1." Again, this is an expected result, because 1 is the slope of the straight line
"The derivative of a sum or difference This follows from Theorem 1 on limits, Lesson 2. For example,
according to the Theorem of Lesson 4, and 1, 2, and 3 above.
"The derivative of a constant times a function
This follows from Theorem 5 on limits, Lesson 2.
Problem 1. Calculate the derivative of 4 To see the answer, pass your mouse over the colored area.
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"The derivative of a product of two functions is equal to the first times the derivative of the second This is the product rule. We will prove it below.
Example. Accepting for the moment that the derivative of sin
Problem 3. Calculate the derivative of 5
5 The power rule
"The derivative of a power of is equal to the product of That is called the power rule. For example,
It is usual to prove the power rule by means of the binomial theorem. See Topic 24 of Precalculus, especially Problem 5. On applying the definition of the derivative, subtracting However, we have seen that the power rule is true when
It seems natural, then, to give a proof by induction; (Topic 26 of Precalculus). The induction hypothesis will be that the power rule is true for
and we must show that it is true for
Now,
according to the product rule and the induction hypothesis;
Therefore, if the power rule is true for
Problem 4. Calculate the derivative of
6 We will see in Lesson 14 that the power rule is valid for any rational exponent Example. The derivative of the square root. See Lesson 29 of Algebra: Rational Exponents. Problem 5. Calculate the derivative of .
Problem 6. Calculate the derivative of
Proof of the product rule To prove the product rule, we will express the difference quotient simply
Then a change in
On multiplying out the right-hand side,
divide by Δ
Now let Δ Therefore, since the limit of a sum is equal to the sum of the limits (Theorem 1 of limits):
This is the product rule.
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