292 DIFFERENTIAL CALCULUS

27.4 Differentiation of a product

Wheny=uv, anduandvare both functions ofx,

then

dy

dx

=u

dv

dx

+v

du

dx

This is known as theproduct rule.

Problem 10. Find the differential coefficient of

y= 3 x^2 sin 2x.

3 x^2 sin 2xis a product of two terms 3x^2 and sin 2x

Letu= 3 x^2 andv=sin 2x

Using the product rule:

dy

dx

= u

dv

dx

+ v

du

dx

↓↓ ↓ ↓

gives:

dy

dx

=(3x^2 )(2 cos 2x)+(sin 2x)(6x)

i.e.

dy

dx

= 6 x^2 cos 2x+ 6 xsin 2x

= 6 x(xcos 2x+sin 2x)

Note that the differential coefficient of a product

isnotobtained by merely differentiating each term

and multiplying the two answers together. The prod-

uct rule formulamustbe used when differentiating

products.

Problem 11. Find the rate of change ofywith

respect toxgiveny= 3

√

xln 2x.

The rate of change ofywith respect toxis given

by

dy

dx

y= 3

√

xln 2x= 3 x

1

(^2) ln 2x, which is a product.
Letu= 3 x
1
(^2) andv=ln 2x
Then
dy
dx
= u
dv
dx

• 
  

  v
  du
  dx
  ↓↓ ↓ ↓

  
  

  (
  3 x
  1
  2
  )(
  1
  x
  )
  +(ln 2x)
  [
  3
  (
  1
  2
  )
  x
  1
  2 −^1
  ]
  = 3 x
  1
  2 −^1 +(ln2x)
  (
  3
  2
  )
  x−
  1
  2
  = 3 x−
  1
  2
  (
  1 +
  1
  2
  ln 2x
  )
  i.e.
  dy
  dx

  
  

  3
  √
  x
  (
  1 +
  1
  2
  ln 2x
  )
  Problem 12. Differentiatey=x^3 cos 3xlnx.
  Letu=x^3 cos 3x(i.e. a product) andv=lnx
  Then
  dy
  dx
  =u
  dv
  dx
  +v
  du
  dx
  where
  du
  dx
  =(x^3 )(−3 sin 3x)+(cos 3x)(3x^2 )
  and
  dv
  dx

  
  

  1
  x
  Hence
  dy
  dx
  =(x^3 cos 3x)
  (
  1
  x
  )
  +(lnx)[− 3 x^3 sin 3x

  
  


• 3 x^2 cos 3x]
  =x^2 cos 3x+ 3 x^2 lnx(cos 3x−xsin 3x)
  i.e.
  dy
  dx
  =x^2 {cos 3x+3lnx(cos 3x−xsin 3x)}
  Problem 13. Determine the rate of change of
  voltage, givenv= 5 tsin 2tvolts whent= 0 .2s.
  Rate of change of voltage=
  dv
  dt
  =(5t)(2 cos 2t)+( sin 2t)(5)
  = 10 tcos 2t+5 sin 2t
  Whent= 0 .2,
  dv
  dt
  =10(0.2) cos 2(0.2)
  +5 sin 2(0.2)
  =2 cos 0. 4 +5 sin 0.4 (where cos 0. 4
  means the cosine of 0.4 radians)
  Hence
  dv
  dt
  =2(0.92106)+5(0.38942)
  = 1. 8421 + 1. 9471 = 3. 7892
  i.e., the rate of change of voltage whent=0.2 s is
  3.79 volts/s, correct to 3 significant figures.