Introduction to differentiation 317

i.e.

dy

dx

=

8

3

x^3 +

12

x^4

Problem 10. Iff(t)= 4 t+

1

√

t^3

findf′(t)

f(t)= 4 t+

1

√

t^3

= 4 t+

1

t

3

2

= 4 t^1 +t−

3

2

Hence, f′(t)=( 4 )( 1 )t^1 −^1 +

(

−

3

2

)

t−

3

2 −^1

= 4 t^0 −

3

2

t−

5

2

i.e. f′(t)= 4 −

3

2 t

5

2

= 4 −

3

2

√

t^5

Problem 11. Determine

dy

dx

giveny=

3 x^2 − 5 x

2 x

y=

3 x^2 − 5 x

2 x

=

3 x^2

2 x

−

5 x

2 x

=

3

2

x−

5

2

Hence,

dy

dx

=

3

2

or1.5

Problem 12. Find the differential coefficient of

y=

2

5

x^3 −

4

x^3

+ 4

√

x^5 + 7

y=

2

5

x^3 −

4

x^3

+ 4

√

x^5 + 7

i.e. y=

2

5

x^3 − 4 x−^3 + 4 x

5

(^2) + 7
dy
dx

(
2
5
)
( 3 )x^3 −^1 −( 4 )(− 3 )x−^3 −^1
+( 4 )
(
5
2
)
x
5
2 −^1 + 0

6
5
x^2 + 12 x−^4 + 10 x
3
2
i.e.
dy
dx

6
5
x^2 +
12
x^4

• 
  

  10
  √
  x^3
  Problem 13. Differentiatey=
  (x+ 2 )^2
  x
  with
  respect tox
  y=
  (x+ 2 )^2
  x

  
  

  x^2 + 4 x+ 4
  x

  
  

  x^2
  x

  
  


• 
  

  4 x
  x

  
  


• 
  

  4
  x
  i.e. y=x^1 + 4 + 4 x−^1
  Hence,
  dy
  dx
  = 1 x^1 −^1 + 0 +( 4 )(− 1 )x−^1 −^1
  =x^0 − 4 x−^2 = 1 −
  4
  x^2
  (sincex^0 = 1 )
  Problem 14. Find the gradient of the curve
  y= 2 x^2 −
  3
  x
  atx= 2
  y= 2 x^2 −
  3
  x
  = 2 x^3 − 3 x−^1
  Gradient=
  dy
  dx
  =( 2 )( 2 )x^2 −^1 −( 3 )(− 1 )x−^1 −^1
  = 4 x+ 3 x−^2
  = 4 x+
  3
  x^2
  Whenx= 2 , gradient= 4 x+
  3
  x^2
  = 4 ( 2 )+
  3
  ( 2 )^2
  = 8 +
  3
  4
  =8.75
  Problem 15. Find the gradient of the curve
  y= 3 x^4 − 2 x^2 + 5 x−2 at the points( 0 ,− 2 )
  and( 1 , 4 )
  The gradient of a curve at a given point is given by the
  corresponding value of the derivative.
  Thus, sincey= 3 x^4 − 2 x^2 + 5 x−2,
  thegradient=
  dy
  dx
  = 12 x^3 − 4 x+ 5.
  At the point( 0 ,− 2 ),x=0, thus
  thegradient= 12 ( 0 )^3 − 4 ( 0 )+ 5 = 5
  At the point( 1 , 4 ),x=1, thus
  thegradient= 12 ( 1 )^3 − 4 ( 1 )+ 5 = 13
  Now try the following Practice Exercise
  PracticeExercise 133 Differentiation of
  y=axnby the general rule (answerson
  page 354)
  In problems 1 to 20, determine the differential
  coefficients with respect to the variable.

  
  

1. y= 7 x^4 2. y= 2 x+ 1


2. y=x^2 −x 4. y= 2 x^3 − 5 x+ 6