Textbook pdf's

IN SUMMARY
Key Ideas
• The derivative of a product of differentiable functions is not the product
of their derivatives.
• The product rule for differentiation:
If h1x2 f1x2g1x2, then h¿1x2 f¿1x2g1x2 f1x2g¿1x2.
• The power of a function rule for integers:
If f1x2 3g1x24 n , then f¿1x2 n3g1x24 n1 g¿1x2.
Need to Know
• In some cases, it is easier to expand and simplify the product before
differentiating, rather than using the product rule.
If
f1x2 3x 4 15x 3 72
15x 7 21x 4
f¿1x2 105x 6 84x 3
• If the derivative is needed at a particular value of the independent variable,
it is not necessary to simplify before substituting.
Exercise 2.3
K
PART A
1. Use the product rule to differentiate each function. Simplify your answers.
a. h1x2 x1x 42
d. h1x2 15x 7 121x 2 2x2
b. h1x2 x 2 12x 12
e. s1t2 1t 2 1213 2t 2 2
c. h1x2 13x 2212x 72 f. f 1x2 x 3
x 3
2. Use the product rule and the power of a function rule to differentiate
the following functions. Do not simplify.
a. y 15x 12 3 1x 42
c. y 11 x 2 2 4 12x 62 3
b. y 13x 2 4213 x 3 2 5 d. y 1x 2 92 4 12x 12 3
3. When is it not appropriate to use the product rule? Give examples.
4. Let F1x2 3b1x243c1x24. Express F1x2 in terms of b1x2 and c1x2.
90
2.3 THE PRODUCT RULE
NEL