Worksheet: Logarithmic Differentiation

In this worksheet, we will practice finding the derivatives of positive functions by taking the natural logarithm of both sides before differentiating.

Q1:

Using logarithmic differentiation, determine the derivative of 𝑦 = 𝑥 + 1 2 𝑥 2 .

  • A 𝑦 = 𝑥 + 1 2 𝑥 2 1 2 𝑥 + 2 8 𝑥 2 𝑥 2
  • B 𝑦 = 𝑥 + 1 2 𝑥 2 1 2 𝑥 + 2 + 4 𝑥 2 𝑥 2
  • C 𝑦 = 𝑥 + 1 2 𝑥 2 1 2 𝑥 + 2 1 2 𝑥 2
  • D 𝑦 = 𝑥 + 1 2 𝑥 2 1 2 𝑥 + 2 2 𝑥 𝑥 1
  • E 𝑦 = 𝑥 + 1 2 𝑥 2 1 2 𝑥 + 2 𝑥 2 𝑥 2

Q2:

Use logarithmic differentiation to find the derivative of the function 𝑦 = 2 ( 𝑥 ) c o s .

  • A 𝑦 = 2 ( 𝑥 ) [ 𝑥 𝑥 𝑥 ] c o s l n c o s c o t
  • B 𝑦 = 2 ( 𝑥 ) [ 𝑥 + 𝑥 𝑥 ] c o s l n c o s t a n
  • C 𝑦 = [ 𝑥 + 𝑥 𝑥 ] [ 𝑥 𝑥 𝑥 ] l n c o s c o t l n c o s t a n
  • D 𝑦 = 2 ( 𝑥 ) [ 𝑥 𝑥 𝑥 ] c o s l n c o s t a n
  • E 𝑦 = 𝑥 𝑥 𝑥 l n c o s t a n

Q3:

Use logarithmic differentiation to find the derivative of the function 𝑦 = 5 𝑥 c o s .

  • A 𝑦 = 2 5 𝑥 𝑥 𝑥 + 𝑥 𝑥 c o s c o s s i n l n
  • B 𝑦 = 𝑥 𝑥 𝑥 𝑥 c o s s i n l n
  • C 𝑦 = 2 5 𝑥 𝑥 𝑥 𝑥 𝑥 c o s c o s s i n l n
  • D 𝑦 = 2 5 𝑥 𝑥 𝑥 𝑥 𝑥 c o s c o s s i n l n
  • E 𝑦 = 2 5 𝑥 𝑥 𝑥 + 𝑥 𝑥 c o s c o s s i n l n

Q4:

Use logarithmic differentiation to find the derivative of the function 𝑦 = 3 ( 𝑥 ) t a n .

  • A 𝑦 = 9 ( 𝑥 ) 4 𝑥 𝑥 𝑥 + ( 𝑥 ) 𝑥 t a n s e c t a n l n t a n
  • B 𝑦 = 9 ( 𝑥 ) 4 𝑥 𝑥 𝑥 + ( 𝑥 ) 𝑥 t a n s e c t a n l n t a n
  • C 𝑦 = 9 ( 𝑥 ) 4 𝑥 𝑥 ( 𝑥 ) 𝑥 t a n s e c l n t a n
  • D 𝑦 = 9 ( 𝑥 ) 4 𝑥 𝑥 𝑥 ( 𝑥 ) 𝑥 t a n s e c t a n l n t a n
  • E 𝑦 = 3 4 𝑥 𝑥 𝑥 ( 𝑥 ) 𝑥 s e c t a n l n t a n

Q5:

Use logarithmic differentiation to find the derivative of the function 𝑦 = ( 𝑥 ) l n c o s .

  • A 𝑦 = 3 ( 𝑥 ) [ 𝑥 𝑥 𝑥 𝑥 ( 𝑥 ) ] l n l n c o s s i n l n l n c o s
  • B 𝑦 = 3 ( 𝑥 ) 𝑥 𝑥 𝑥 + 𝑥 ( 𝑥 ) l n c o s l n s i n l n l n c o s
  • C 𝑦 = ( 𝑥 ) 3 [ 𝑥 𝑥 𝑥 𝑥 ( 𝑥 ) ] l n l n c o s s i n l n l n c o s
  • D 𝑦 = 3 ( 𝑥 ) 𝑥 𝑥 𝑥 𝑥 ( 𝑥 ) l n c o s l n s i n l n l n c o s
  • E 𝑦 = 3 𝑥 𝑥 𝑥 𝑥 ( 𝑥 ) c o s l n s i n l n l n

Q6:

If 5 𝑦 = 3 𝑥 , determine d d 𝑦 𝑥 .

  • A 1 8 5 𝑥
  • B 1 8 𝑥 ( 3 𝑥 ) l n
  • C 3 5 𝑥
  • D 1 8 5 𝑥 ( 𝑥 + 1 ) l n

Q7:

Determine d d 𝑦 𝑥 , given that 𝑦 = 5 𝑥 + 1 1 .

  • A 5 𝑥 + 1 1 5 𝑥 + 1 5 𝑥 5 𝑥 + 1 1 l n
  • B 1 5 𝑥 5 𝑥 + 1 1 ( 5 𝑥 + 1 1 ) l n
  • C 3 5 𝑥 + 1 1 + 1 5 𝑥 5 𝑥 + 1 1 l n
  • D 5 𝑥 + 1 1 3 5 𝑥 + 1 1 + 6 0 𝑥 5 𝑥 + 1 1 l n
  • E l n 5 𝑥 + 1 1 𝑥 ( 5 𝑥 + 1 1 )

Q8:

Find d d 𝑦 𝑥 if 𝑦 = 6 𝑥 + 7 .

  • A 8 𝑦 6 𝑥 + 7 + 5 4 𝑥 l n
  • B 8 6 𝑥 + 7 + 5 4 𝑥 6 𝑥 + 7 l n
  • C 8 𝑦 6 𝑥 + 7 + 5 4 𝑥 6 𝑥 + 7 l n
  • D 8 𝑦 6 𝑥 + 7 + 5 4 𝑥 6 𝑥 + 7 l n

Q9:

Determine d d 𝑦 𝑥 , given that 𝑦 = ( 8 4 𝑥 ) s i n .

  • A 1 6 𝑥 4 𝑥 c o s
  • B ( 8 4 𝑥 ) [ ( 4 𝑥 ) + 𝑥 4 𝑥 ] s i n l n s i n t a n
  • C 1 6 𝑥 ( 4 𝑥 ) l n c o s
  • D 2 ( 8 4 𝑥 ) [ ( 8 4 𝑥 ) + 4 𝑥 4 𝑥 ] s i n l n s i n c o t

Q10:

Determine d d 𝑦 𝑥 , if 𝑦 = ( 5 4 𝑥 ) s i n t a n .

  • A t a n l n c o s 4 𝑥 ( 2 0 4 𝑥 )
  • B 1 + 4 𝑥 ( 5 4 𝑥 ) s e c l n s i n
  • C 2 0 4 𝑥 4 𝑥 + 2 0 4 𝑥 4 𝑥 s i n s e c c o s t a n
  • D 4 ( 5 4 𝑥 ) 1 + 4 𝑥 ( 5 4 𝑥 ) s i n s e c l n s i n t a n

Q11:

Determine d d 𝑦 𝑥 for the function 𝑦 𝑦 = ( 3 𝑥 + 8 ) : c o s .

  • A [ 3 𝑥 + 8 ] 1 6 ( 3 𝑥 + 8 ) 8 𝑥 + 6 8 𝑥 3 𝑥 + 8 c o s l n s i n c o s
  • B 1 6 ( 3 𝑥 + 8 ) 8 𝑥 6 8 𝑥 3 𝑥 + 8 l n s i n c o s
  • C [ 3 𝑥 + 8 ] 1 6 ( 3 𝑥 + 8 ) 8 𝑥 + 6 8 𝑥 3 𝑥 + 8 c o s l n s i n c o s
  • D [ 3 𝑥 + 8 ] 1 6 ( 3 𝑥 + 8 ) 8 𝑥 6 8 𝑥 3 𝑥 + 8 c o s l n s i n c o s

Q12:

Find d d 𝑦 𝑥 , given that 7 𝑦 = 6 𝑥 s i n .

  • A 6 7 𝑥 ( 𝑥 6 𝑥 + 6 6 𝑥 𝑥 ) s i n s i n c o s l n
  • B 3 6 7 𝑥 6 𝑥 𝑥 + 6 𝑥 6 𝑥 s i n s i n l n c o s
  • C s i n c o s l n 6 𝑥 𝑥 + 6 6 𝑥 𝑥
  • D 6 7 𝑥 6 𝑥 𝑥 + 6 𝑥 6 𝑥 s i n s i n l n c o s
  • E 6 7 𝑥 6 𝑥 𝑥 6 𝑥 6 𝑥 s i n s i n l n c o s

Q13:

Find d d 𝑦 𝑥 if 6 𝑦 = 7 𝑥 .

  • A 7 1 0 𝑥 ( 1 𝑥 ) l n
  • B 7 1 0 𝑥 ( 1 𝑥 ) l n
  • C 7 1 0 𝑥
  • D 7 1 0 𝑥 ( 1 𝑥 ) l n
  • E 3 5 𝑥 ( 1 𝑥 ) l n

Q14:

Given that 𝑦 = ( 8 𝑥 ) l o g t a n , find d d 𝑦 𝑥 .

  • A 2 0 ( 8 𝑥 ) 5 𝑥 + 4 5 𝑥 𝑥 8 𝑥 l n l o g s e c t a n l o g
  • B 2 0 ( 8 𝑥 ) 5 𝑥 + 4 5 𝑥 𝑥 1 0 8 𝑥 l n l o g s e c t a n l n l o g
  • C ( 8 𝑥 ) 2 0 ( 8 𝑥 ) 5 𝑥 + 4 5 𝑥 𝑥 1 0 l o g l n l o g s e c t a n l n t a n
  • D ( 8 𝑥 ) 2 0 ( 8 𝑥 ) 5 𝑥 + 4 5 𝑥 𝑥 1 0 8 𝑥 l o g l n l o g s e c t a n l n l o g t a n

Q15:

Given that 𝑦 = 2 s i n , determine d d 𝑦 𝑥 .

  • A 2 8 1 𝑒 𝑥 2 s i n c o s l n
  • B 2 8 1 𝑒 + 𝑥 s i n c o s
  • C 9 𝑒 + 𝑥 2 c o s l n
  • D 2 8 1 𝑒 + 𝑥 2 s i n c o s l n

Q16:

Given that 𝑦 = ( 3 5 𝑥 ) l o g l o g , find d d 𝑦 𝑥 .

  • A 1 𝑥 1 0 ( ( 3 5 𝑥 ) + 1 ) l n l n l o g
  • B 𝑦 1 + ( 3 5 𝑥 ) 𝑥 1 0 l n l o g l n
  • C 1 + ( 3 5 𝑥 ) 𝑥 1 0 l n l o g l n
  • D 𝑦 𝑥 1 0 ( ( 3 5 𝑥 ) + 1 ) l n l n l o g

Q17:

Given 𝑦 = 𝑥 𝑥 𝑥 , find d d 𝑦 𝑥 .

  • A l n l n l n 𝑦 𝑥 + 1 𝑥 𝑥 + 1
  • B 𝑦 𝑦 1 𝑥 𝑥 + 1 l n l n
  • C 𝑦 𝑦 𝑥 + 1 𝑥 + 1 l n l n l n
  • D 𝑦 𝑦 𝑥 + 1 𝑥 𝑥 + 1 l n l n l n

Q18:

If 𝑦 = 𝑒 𝑥 + 4 𝑥 + 4 , determine 1 6 𝑥 𝑦 .

  • A 𝑦 6 𝑥 2 0
  • B 𝑦 6 𝑥 9 2
  • C 6 𝑥 𝑦
  • D 𝑦 6 𝑥 9 2
  • E 6 𝑥 + 4 𝑦

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