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.
"The derivative of a constant is 0."
We should expect this, because the slope of a horizontal line y = c is 0.
"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 y = x. (Topic 9 of Precalculus.)
"The derivative of a sum or difference
This follows from Theorem 1 on limits, Lesson 2.
"The derivative of a constant times a function
This follows from Theorem 5 on limits, Lesson 2.
Problem 1. Calculate the derivative of 4x² − 6x + 2.
To see the answer, pass your mouse over the colored area.
8x − 6.
"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 x is cos x (Lesson 12), then
Problem 3. Calculate the derivative of 5x sin x.
5x cos x + 5 sin x
The power rule
"The derivative of a power of x
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 xn, dividing the numerator by h and taking the limit, the rule follows.
However, we have seen that the power rule is true when n = 1:
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 n = k:
and we must show that it is true for n = k + 1; i.e. that
Therefore, if the power rule is true for n = k, then it is also true for its successor, k + 1. And since the rule is true for n = 1, it is therefore true for every natural number.
Problem 4. Calculate the derivative of x6 − 3x4 + 5x3 − x + 4.
6x5 − 12x3 + 15x² − 1
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 x.
Proof of the product rule
To prove the product rule, we will express the difference quotient simply
y = f g.
Then a change in y -- Δy -- will produce corresponding changes in f and g:
y + Δy = (f + Δf )(g + Δg)
On multiplying out the right-hand side,
y + Δy = f g + f Δg + g Δf + Δf Δg.
divide by Δx:
Now let Δx0. Hence Δy will approach 0, as will both Δf and Δg, so that the last term on the right approaches 0.
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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