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 = x5 and 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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