Example 13: If y = 3x^4 + 8x^10 , then = 12x^3 + 80x^9.
Example 14: If y = 7x−4 + 5x
−
, then = −28x−5 − x−
.Example 15: If y = 5x^4 (2 − x^3 ), then = 40x^3 − 35x^6.
Example 16: If y = (3x^2 + 5)(x − 1), then
y = 3x^3 − 3x^2 + 5x − 5 and = 9x^2 − 6x + 5.Example 17: If y = ax^3 + bx^2 + cx + d, then = 3ax^2 + 2bx + c.
After you’ve worked through all 17 of these examples, you should be able to take the derivative of any
polynomial with ease.
As you may have noticed from the examples above, in calculus, you are often asked to convert from
fractions and radicals to negative powers and fractional powers. In addition, don’t freak out if your
answer doesn’t match any of the answer choices. Because answers to problems are often presented in
simplified form, your answer may not be simplified enough.
There are two basic expressions that you’ll often be asked to differentiate. You can make your life easier
by memorizing the following derivatives:
If y = , then = −If y = k , then = This first formula is also known as the “Reciprocal Rule.”HIGHER ORDER DERIVATIVES
This may sound like a big deal, but it isn’t. This term refers only to taking the derivative of a function
more than once. You don’t have to stop at the first derivative of a function; you can keep taking
derivatives. The derivative of a first derivative is called the second derivative. The derivative of the
second derivative is called the third derivative, and so on.