If f(x) = f(x)g(x), where f and g have derivatives of all orders
(a) show that f ′′ = f ′′g + 2f ′ g ′ + fg′′ .
(b) find similar formulas for f ′′′ and f (4) .
(c) describe the pattern for higher derivatives of f.

a) The second derivative is .

b) The third and fourth derivatives are and ,  respectively.

c) There is a power binomial pattern for higher derivatives of F.

In this question we are going to derive expressions for by derivative rules, to be more specific, the derivative rule for the product between two functions.

a) By the derivative rule for the product between two functions we have the following result:

First derivative

Second derivative

(1)

The second derivative is .

b) And we find the formulas for the third and fourth derivatives of the expression from (1):

Third derivative

(2)

Fourth derivative

(3)

The third and fourth derivatives are and ,  respectively.

c) Based on (1), (2) and (3), we notice a power binomial pattern defined by the power binomial theorem. In other words, we can derive this expression by analogy:

(4)

Where:

There is a power binomial pattern for higher derivatives of F.

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a. By the product rule,

b. By the same rule,

and

c. You might recognize the coefficients as those that appear in the expansion of :

1, 1

1, 2, 1

1, 3, 3, 1

1, 4, 6, 4, 1

and so on; the general pattern (known as the general Leibniz rule) is

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by definition, an arc subtended by a central angle of some given measure has the same measure. so .