CHAPTER 8
The Derivative
lim
8.1
8.2
Using the A-definition, find the derivative /'(x) of the function /(x) = 2x - 7.
Hence
Thus,
Answer
Using the A-definition, show that the derivative of any linear function /(x) = Ax + B is f'(x) = A.
Hence.
Thus,
8.3
Using the A-definition, find the derivative f'(x) of the function /(x) = 2x
2 - 3x + 5.
Thus,
Hence,
49
So,
Then,
8.9
Using the formula from Problem 8-7. find the derivative of
8.8
Using the product rule, find the derivative of f(x) = (Sx
3 - 20* + 13)(4;t
6 + 2x
5 - lx
2 + 2x).
F'(x) = (5x
3 -2Qx+13)(24x
5 + Wx
4 -Ux + 2) + (4x" + 2x
5 -Ix
2 + 2x)(15x
2 -20). [In such cases, do
not bother to carry out the tedious multiplications, unless a particular problem requires it.]
[Notice the various ways of denoting a derivative:
Given functions f(x) and g(x), state the formulas for the derivatives of the sum f(x) + g(x), the product
fix) • e(x), and the quotient f(x) /g(x).
8.7
8.6
Write the derivative of the function f(x) = lx~ - 3x
4 + 6x
2 + 3x + 4.
/'(*) = 35x
4 - I2x
3 + \2x + 3.
State the formula for the derivative of an arbitrary polynomial function f(x) = a n x" + a n _ l x" ' + ••• + a 2 x
2 +
8.5
8.4
Using the A-definition, find the derivative /'(*) of the function f(x) = x
3 .
So,
So,
Thus,
a1x+a0.