DIFFERENTIATION USING PRODUCT RULE

The product rule is the formal rule for differentiating problems where one function is multiplied by another.

(UV)' = UV' + VU'

Example 1 :

Differentiate x5 tan x

Solution :

Let y  =  x5 tan x

u  =  xand u'  =  5x4 

v = tan x and v'  =  sec2 x

Using the formula 

(UV)' = UV' + VU'

=  (x5)sec2x+(tan x)(5x4)

x5sec2x+5x4tan x

=  x4[xsec2x+tanx]

Example 2 :

Differentiate (x2+7x+2) (x3-log x)

Solution :

u  =  x2+7x+2 and  u' = 2x+7

v  =  x3 - log x and v' = 3x2 - (1/x) 

Formula of product rule

           = (x2+7x+2)[3x2 - (1/x)] + (x3 - log x)(2x+7)

Example 3 :

Differentiate (x2-1) (x2+2)

Solution :

u  =  x2-1 and u'  =  2x

v = x2+2 and v'  =  2x

Using the formula, we get

(UV)' = UV' + VU'

=  (x2-1)(2x)+(x2+2)(2x)

=  2x3-2x+2x3+4x

=  4x3+2x

Another Method :

(x2-1) (x2+2)

=  x2(x2) + 2x2 - 1(x2) - 1(2)

=  x4+2x2-x2-2

=  x4+x2-2

Differentiating

=  4x3 + 2x - 0

=  4x³ + 2x

Example 4 :

If f(2)  =  −8, f′(2)  =  3, g(2)  =  17 and g′(2)  =  −4 

determine the value of (fg)′(2).

Solution :

(fg)'(x)  =  f(x) g'(x)+f'(x)g(x)

(fg)'(2)  =  f(2) g'(2)+f'(2)g(2)

  =  -8(-4) + 17(3)

  =  32 + 51

  =  83

Example 5 :

Suppose that f and g are functions that are differentiable at x = 1 and that 

f(1)  =  2, f′(1)  =  −1, g(1)  =  −2, and g′(1)  =  3. Find h′(1).

If h(x)  =  (x2+9)g(x)

Solution :

Given that :

h(x)  =  (x2+9)g(x)

Let f(x)  =  x2+9

h'(x)  = (x2+9)g'(x) + g(x) d(x2+9)

h'(x)  =  (x2+9)g'(x) + g(x) (2x)

h'(1)  =  (12+9)g'(1) + g(1) 2(1)

  =  10(3)+(-2)2

  =  30 - 4

  =  26

Example 6 :

If

f(x)  =  x3g(x), g(−7)  =  2, g′(−7)  =  −9 

determine the value of f′(−7). 

Solution :

Given that :

f(x)  =  x3g(x)

f'(x)  =  x3g'(x) + g(x) d(x3)

f'(x)  =  x3g'(x) + 3x2g(x)

f'(-7)  =  (-7)3g'(-7) + 3(-7)2g(-7)

f'(-7)  =  -343(2) + 3(49)(-9)

f'(-7)   =  -686+1323

f'(-7)  =  637

Example 7 :

Suppose that

f(π/4)  =  −4 and  f′(π/4)  =  2, and let  g(x)  =  f(x) sinx.

Solution :

g(x)  =  f(x) sinx

g'(x)  =  f(x) d(sinx) + sinx f'(x)

g'(x)  =  f(x) cosx + sinx f'(x)

g'(π/4)  =  f(π/4) cosπ/4 + sinπ/4 f'(π/4)

=  -4(1/√2) + (1/√2)2

=  -2/√2

=  -√2

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