Product rule, Chain rule and Higher derivatives
Bridge Course, August 2012

Product Rule
     Intuition: The derivative of a product is not the product of the derivatives. That is, differentiation does not distribute over multiplication. To find the derivative of a product expression, we need to use product rule.
     Definition: In calculus, the product rule is a formula used to find the derivatives of products of functions. It is defined as

or written in another format:

There is another rule for quotients.That is quotient rule and it is defined as

Examples                                                                                                 MAXIMA CODE

1.   Find .

Consider  and
 and  and hence,

diff((x ^ 2 + 2 * x + 3) * (x ^ 3 + 2 * x − 2), x, 1)

2.  Find .

Consider  and
 and  and hence,

                                          diff(%e ^ (3 * x) * tan(x), x, 1)

Exercises
Find the derivative of the following function

                                                                  diff((2 * x − 1) * (x + 1) ^ 2, x, 1)

                                                                       diff(%e ^ x * cos(x) * sin(x), x, 1)

                                                                                   diff(x * log(x), x, 1)

                                                                                     diff(cos(x)/sin(x), x, 1)

                                                             diff(%e ^ x * tan(x)/x, x, 1)

Chain Rule
Intuition: The chain rule states that if we have a function of the form y(u(x)) (i.e. y can be written as a function of u and u can be written as a function of x) then

Definition: If a function F(x) is composed to two differentiable functions g(x) and m(x), so that F(x) = g(m(x)), then F(x) is differentiable and,

Examples

1.  Compute , where                                       diff((1 + 3 * x^2)(1/2), x, 1)

Let  and , so  and .

Then, by applying the chain rule, we have

2. Compute , where                               diff(sin(cos(x)), x, 1)

Let , so  and .Then, by chain rule, we have

Exercises
Find the derivatives of the following function

1.                                                                          diff((x ^ 2+ 10) ^ (1/3), x, 1)
2.                                                                     diff(x ^ 2 * log(1 − 2 * x), x, 1)
3.                                                                          diff(x/sqrt(1 − 2 * x^2), x, 1)
4.                                                                      diff(tan(sqrt(2 − 3 * x)), x, 1)
5.                                                                               diff(%e ^ (−sin(1/x ^2)), x, 1)
6.                                                                         diff(log((1 − x ^ 2) ^ 2), x, 1)
7.                                                                                       diff(%e ^ (%e ^ (2 ∗ x)), x, 1)
8.                                                                           diff(atan(3 * x ^2), x, 1)

Higher derivatives
Definition: Let f be a differentiable function, and let f′(x) be its derivative. The derivative of f′(x) (if it exists) is written f′′(x) and is called the second derivative of f. Similarly, the derivative of a second derivative, if it exists, is written f′′′(x) and is called the third derivative of f. These repeated derivatives are called higher-order derivatives.

Examples

1.   Find the third derivative of  with respect to x.

diff(x ^ 5 + 6 * x ^ 3 + 4 * x + 3, x, 3)

2.  Find the third derivative of  with respect to x.

diff(12 * sin(2 * x) + log(x + 2) + 2 * x, x, 3)

Exercises

1.  Find the third derivative of                          diff((100 * x + 1) ^ 2, x, 3)
2.  Find the third derivative of                                     diff(sin(x), x, 3)
3.  Find the third derivative of                                    diff(tan(x), x, 3)
4.  Find the third and -th derivative of                           find it yourself

5.  Find the third and -th derivative of                find it yourself