If y = y(v) and v = v(x), then As you can see, these grow quite complex, so we simplify
these only as a last resort. If you must simplify, the AP Exam
will have only a very simple Chain Rule problem.Example 12: y = 8v^2 − 6v and v = 5x^3 − 11x, then
=(16v − 6)(15x^2 − 11)Then substitute for v.
= (16(5x^3 − 11x) − 6)(15x^2 − 11) = (80x^3 − 176x − 6)(15x^2 − 11)Here are some solved problems. Cover the answers first, then check your work.
PROBLEM 1. Find if y = (5x^4 + 3x^7 )(x^10 − 8x).
Answer: if y = (5x^4 + 3x^7 )(10x^9 − 8)(x^10 − 8x)(20x^3 + 21x^6 ).
PROBLEM 2. Find if y = (x^3 + 3x^2 + 3x + 1)(x^2 + 2x + 1).
Answer: = (x^3 + 3x^2 + 3x + 1)(2x + 2) + (x^2 + 2x + 1)(3x^2 + 6x + 3)
PROBLEM 3. Find if y = .
Answer:
PROBLEM 4. Find if y = (x^3 + 1)(x^2 + 5x − ).
Answer: = (x^3 + 1)(2x + 5 + ) + (x^2 + 5x − )(3x^2 )