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4 Product rule Chain rule and Higher derivatives

Page history last edited by CAI, Sheng 11 years, 10 months ago

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

               Formula

or written in another format:

                                   Formula

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

                         Formula

 

Examples                                                                                                 MAXIMA CODE

1.   Formula Find Formula.

 

Consider Formula and Formula
Formula and Formula and hence,

Formula

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

 

 

2.  Formula Find Formula.

 

Consider Formula and Formula
Formula and Formula and hence,

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

 

Exercises
Find the derivative of the following function

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

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

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

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

Formula                                                             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

                                                                    Formula

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,

                                                            Formula

 

Examples

 

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

 

Let Formula and Formula, so Formula and Formula.

Then, by applying the chain rule, we have

               Formula

 

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

 

Let Formula, so Formula and Formula.Then, by chain rule, we haveFormula

 

Exercises
Find the derivatives of the following function

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

 

                    Formula

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

 

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

                    Formula

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

 

Exercises

 

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

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

 

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