11DIFFERENTIATION - Department of Mathematics

SOLUTION ✔
Rule 6: The Quotient Rule
11.2 THE PRODUCT AND QUOTIENT RULES 709
First, we express the function in exponential form, obtaining
f(x) x 3 (x 1/2 1)
By the product rule,
3 d
f(x) x
dx (x1/2 1) (x 1/2 1) d 3 x
dx
x 31 2 x1/2 (x 1/2 1)(3x 2 )
1
2 x 5/2 3x 5/2 3x 2
7
2 x 5/2 3x 2
REMARK We can also solve the problem by first expanding the product
before differentiating f. Examples for which this is not possible will be considered
in Section 11.3, where the true value of the product rule will be appreciated.
T HE Q UOTIENT R ULE
The derivative of the quotient of two differentiable functions is given by the
following rule:
d
dx f(x) g(x)f(x) f(x)g(x)
g(x)
[g(x)] 2
(g(x) 0)
As an aid to remembering this expression, observe that it has the following
form:
d
dx f(x)
g(x)
Derivative of
Derivative of
(Denominator)
numerator (Numerator)
denominator
(Square of denominator)
For a proof of the quotient rule, see Exercise 64, page 717.
The derivative of a quotient is not equal to the quotient of the derivatives;
that is,
d
dx f(x)
g(x) f(x)
g(x)