Worksheet: Directional Derivatives and Gradient

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In this worksheet, we will practice finding a derivative of multivariable functions in a given direction (directional derivative) and finding the gradient vector of the function.

Q1:

Find the directional derivative of 𝑓(𝑥,𝑦)=𝑥+𝑦−1 at the point (1,1) in the direction of 𝑣=1√2,1√2.

  • A √ 2
  • B 3 √ 2
  • C 4 √ 2
  • D4
  • E 2 √ 2

Q2:

Find the directional derivative of 𝑓(𝑥,𝑦)=𝑥𝑒 at the point (1,1) in the direction of 𝑣=1√2,1√2.

  • A 2 𝑒 √ 2
  • B 2 √ 2 𝑒
  • C 2 𝑒 √ 2
  • D 3 𝑒 √ 2
  • E 3 𝑒 √ 2

Q3:

Find the directional derivative of 𝑓(𝑥,𝑦,𝑧)=𝑥𝑒 at the point (1,1,1) in the direction of 𝑣=1√3,1√3,1√3.

  • A 2 𝑒 √ 3
  • B 4 𝑒
  • C 3 𝑒 √ 3
  • D 4 𝑒 √ 3
  • E 𝑒 √ 3

Q4:

Find the directional derivative of 𝑓(𝑥,𝑦,𝑧)=𝑥𝑦𝑧sin at the point (1,1,1) in the direction of v=1√3,1√3,1√3.

  • A 3 √ 3 1 c o s
  • B c o s 1 √ 3
  • C √ 3 1 c o s
  • D √ 3
  • E 3 √ 3

Q5:

Find the directional derivative of 𝑓(𝑥,𝑦)=1𝑥+𝑦 at the point (1,1) in the direction of 𝑣=1√2,1√2.

  • A √ 2
  • B √ 2 2
  • C − √ 2 2
  • D − √ 2
  • E − 3 √ 2 2

Q6:

Find the directional derivative of 𝑓(𝑥,𝑦)=√𝑥+𝑦+4 at the point (1,1) in the direction of 𝑣=1√2,1√2.

  • A 2 √ 3 3
  • B √ 3 3
  • C √ 3
  • D 3 √ 2 2
  • E √ 2 2

Q7:

Compute the gradient of 𝑓(𝑥,𝑦)=√𝑥+𝑦+4.

  • A 𝑥 √ 𝑥 + 𝑦 + 4 , 𝑦 √ 𝑥 + 𝑦 + 4
  • B 𝑥 √ 𝑥 + 𝑦 + 4 , 𝑦 √ 𝑥 + 𝑦 + 4
  • C 2 𝑥 √ 𝑥 + 𝑦 + 4 , 2 𝑦 √ 𝑥 + 𝑦 + 4
  • D 2 𝑦 √ 𝑥 + 𝑦 + 4 , 2 𝑥 √ 𝑥 + 𝑦 + 4
  • E 𝑦 √ 𝑥 + 𝑦 + 4 , 𝑥 √ 𝑥 + 𝑦 + 4

Q8:

The temperature 𝑇 of a solid is given by the function 𝑇(𝑥,𝑦,𝑧)=𝑒+𝑒+𝑒, where 𝑥, 𝑦, 𝑧 are space coordinates relative to the center of the solid. In which direction from the point (3,1,2) will the temperature decrease the fastest?

  • A 𝑇 decreases the fastest in the direction of −𝑒,−2𝑒,4𝑒.
  • B 𝑇 decreases the fastest in the direction of 𝑒,2𝑒,−4𝑒.
  • C 𝑇 decreases the fastest in the direction of 𝑒,2𝑒,−4𝑒.
  • D 𝑇 decreases the fastest in the direction of −𝑒,−2𝑒,4𝑒.

Q9:

In which direction does the function 𝑓(𝑥,𝑦)=𝑥𝑦+𝑥𝑦 increases the fastest from the point (2,3)? In which direction does it decrease the fastest? Give your answer using unit vectors.

  • A 𝑓 increases the fastest in the direction of 29√941,10√941 and decreases the fastest in the direction of −29√941,−10√941.
  • B 𝑓 increases the fastest in the direction of −29√941,−10√941 and decreases the fastest in the direction of 29√941,10√941.
  • C 𝑓 increases the fastest in the direction of 9√97,4√97 and decreases the fastest in the direction of −9√97,−4√97.
  • D 𝑓 increases the fastest in the direction of −9√97,−4√97 and decreases the fastest in the direction of 9√97,4√97.

Q10:

Consider the function 𝑓ℝ→ℝ:, given by 𝑓(𝑥,𝑦)=𝑥𝑦sin. Which of the statements below is incorrect?

  • A 𝐷 𝑓 ( 𝑥 , 𝑦 ) = 𝐷 𝑓 ( 𝑥 , 𝑦 ) = 0
  • B 𝐷 𝑓 ( 𝑥 , 𝑦 ) = 𝑦 s i n
  • C 𝐷 𝑓 ( 𝑥 , 𝑦 ) = 𝐷 𝑓 ( 𝑥 , 𝑦 )
  • D ( 𝐷 𝑓 ( 𝑥 , 𝑦 ) ) + ( 𝐷 𝑓 ( 𝑥 , 𝑦 ) ) = 1

Q11:

Compute the gradient of the function 𝑓(𝑥,𝑦)=2𝑥+5𝑦.

  • A 1 5 , 1 2
  • B 1 2 , 1 5
  • C ⟨ 2 𝑥 , 5 𝑦 ⟩
  • D ⟨ 2 , 5 ⟩
  • E ⟨ 5 , 2 ⟩

Q12:

Find the gradient of 𝑓(𝑥,𝑦)=𝑥+𝑦−1.

  • A ⟨ 𝑥 , 𝑦 ⟩
  • B ⟨ 2 𝑦 − 1 , 2 𝑥 − 1 ⟩
  • C ⟨ 2 𝑥 − 1 , 2 𝑦 − 1 ⟩
  • D ⟨ 2 𝑦 , 2 𝑥 ⟩
  • E ⟨ 2 𝑥 , 2 𝑦 ⟩

Q13:

Find the gradient of 𝑓(𝑥,𝑦)=𝑥𝑦.ln

  • A 1 𝑥 𝑦 , 1 𝑥 𝑦
  • B 1 𝑥 , 1 𝑦
  • C ( 𝑥 , 𝑦 ) l n l n
  • D ( 𝑥 , 𝑦 )
  • E 1 𝑦 , 1 𝑥

Q14:

Compute the gradient of the function 𝑓(𝑥,𝑦)=𝑥𝑒.

  • A ⟨ 2 𝑒 , 𝑥 𝑒 ⟩
  • B 2 𝑥 𝑒 , 𝑦 𝑥 𝑒
  • C 2 𝑥 𝑒 , 𝑥 𝑒
  • D 𝑥 𝑒 , 2 𝑥 𝑒
  • E ⟨ 𝑥 𝑒 , 2 𝑒 ⟩

Q15:

Find the gradient of 𝑓(𝑥,𝑦)=1𝑥+𝑦.

  • A 2 𝑥 ( 𝑥 + 𝑦 ) , 2 𝑦 ( 𝑥 + 𝑦 )
  • B − 2 𝑥 ( 𝑥 + 𝑦 ) , − 2 𝑦 ( 𝑥 + 𝑦 )
  • C − 𝑥 ( 𝑥 + 𝑦 ) , − 𝑦 ( 𝑥 + 𝑦 )
  • D 2 𝑥 ( 𝑥 + 𝑦 ) , − 2 𝑦 ( 𝑥 + 𝑦 )
  • E − 2 𝑥 ( 𝑥 + 𝑦 ) , 2 𝑦 ( 𝑥 + 𝑦 )

Q16:

Compute the gradient for 𝑓(𝑥,𝑦,𝑧)=𝑥+𝑦+𝑧.

  • A ⟨ 2 𝑥 , 2 𝑧 , 2 𝑦 ⟩
  • B ⟨ 2 𝑦 , 2 𝑥 , 2 𝑧 ⟩
  • C ⟨ 2 , 2 , 2 ⟩
  • D ⟨ 𝑥 , 𝑦 , 𝑧 ⟩
  • E ⟨ 2 𝑥 , 2 𝑦 , 2 𝑧 ⟩

Q17:

Compute the gradient for 𝑓(𝑥,𝑦,𝑧)=𝑥𝑒.

  • A 2 𝑥 𝑒 , 𝑥 𝑧 𝑒 , 𝑥 𝑦 𝑒
  • B 𝑥 𝑧 𝑒 , 2 𝑥 𝑒 , 𝑥 𝑦 𝑒
  • C 𝑥 𝑧 𝑒 , 𝑥 𝑦 𝑒 , 2 𝑥 𝑒
  • D 𝑥 𝑦 𝑒 , 𝑥 𝑧 𝑒 , 2 𝑥 𝑒
  • E 2 𝑦 𝑒 , 𝑦 𝑧 𝑒 , 𝑦 𝑒

Q18:

Compute the gradient for 𝑓(𝑥,𝑦,𝑧)=𝑥𝑦𝑧.sin

  • A ⟨ 𝑦 𝑧 𝑥 𝑦 𝑧 , 𝑥 𝑧 𝑥 𝑦 𝑧 , 𝑥 𝑦 𝑥 𝑦 𝑧 ⟩ c o s c o s c o s
  • B ⟨ 𝑦 𝑧 , 𝑥 𝑧 , 𝑥 𝑦 ⟩
  • C ⟨ 𝑥 𝑧 , 𝑦 𝑧 , 𝑥 𝑦 ⟩
  • D ⟨ 𝑦 𝑧 𝑥 𝑦 𝑧 , 𝑥 𝑦 𝑥 𝑦 𝑧 , 𝑥 𝑧 𝑥 𝑦 𝑧 ⟩ c o s c o s c o s
  • E ⟨ 𝑥 𝑧 𝑥 𝑦 𝑧 , 𝑦 𝑧 𝑥 𝑦 𝑧 , 𝑥 𝑦 𝑥 𝑦 𝑧 ⟩ c o s c o s c o s

Q19:

Compute the gradient for 𝑓(𝑥,𝑦,𝑧)=√𝑥+𝑦+𝑧.

  • A 𝑥 𝑥 + 𝑦 + 𝑧 , 𝑦 𝑥 + 𝑦 + 𝑧 , 𝑧 𝑥 + 𝑦 + 𝑧
  • B 𝑦 √ 𝑥 + 𝑦 + 𝑧 , 𝑥 √ 𝑥 + 𝑦 + 𝑧 , 𝑧 √ 𝑥 + 𝑦 + 𝑧
  • C 𝑥 √ 𝑥 + 𝑦 + 𝑧 , 𝑧 √ 𝑥 + 𝑦 + 𝑧 , 𝑦 √ 𝑥 + 𝑦 + 𝑧
  • D 1 √ 𝑥 + 𝑦 + 𝑧 , 1 √ 𝑥 + 𝑦 + 𝑧 , 1 √ 𝑥 + 𝑦 + 𝑧
  • E 𝑥 √ 𝑥 + 𝑦 + 𝑧 , 𝑦 √ 𝑥 + 𝑦 + 𝑧 , 𝑧 √ 𝑥 + 𝑦 + 𝑧

Q20:

Find a function 𝑄(𝑥,𝑦) so that the vector field F(𝑥,𝑦)=⟨𝑥𝑦,𝑄(𝑥,𝑦)⟩ is a gradient field.

  • A 𝑄 ( 𝑥 , 𝑦 ) = 𝑥 𝑦
  • B 𝑄 ( 𝑥 , 𝑦 ) = 𝑥
  • C 𝑄 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 2
  • D 𝑄 ( 𝑥 , 𝑦 ) = 𝑥 2
  • E 𝑄 ( 𝑥 , 𝑦 ) = 𝑦 2

Q21:

Suppose 𝑤=𝐹(𝜙(𝑥,𝑦)) with 𝜙=(𝑥+𝑦,𝑥−𝑦,𝑥𝑦). Express the gradient ∇𝑤𝜋,−23 (viewed as a 1×2 matrix) in terms of the 1×3 matrix ∇𝐹(𝑞), where 𝑞=𝜙𝜋,−23, and a matrix of partial derivatives of 𝜙.

  • A ∇ 𝑤 𝜋 , − 2 3 = ∇ 𝐹 ( 𝑞 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2 𝜋 4 3 2 𝜋 − 4 3 − 2 3 𝜋 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • B ∇ 𝑤 𝜋 , − 2 3 = ∇ 𝐹 ( 𝑞 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2 𝜋 − 4 3 2 𝜋 − 4 3 − 2 3 𝜋 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • C ∇ 𝑤 𝜋 , − 2 3 = ∇ 𝐹 ( 𝑞 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 2 𝜋 − 4 3 2 𝜋 4 3 − 2 3 𝜋 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • D ∇ 𝑤 𝜋 , − 2 3 = ∇ 𝐹 ( 𝑞 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜋 − 2 3 𝜋 2 3 − 2 3 𝜋 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • E ∇ 𝑤 𝜋 , − 2 3 = ∇ 𝐹 ( 𝑞 ) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜋 − 2 3 𝜋 − 2 3 − 2 3 𝜋 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Q22:

Suppose 𝑤=𝐹(𝜙(𝑥,𝑦)) with 𝜙=(𝜙,𝜙,𝜙) and that 𝑞=𝜙(𝑝) for a point 𝑝∈ℝ. Express the gradient ∇𝑤(𝑝) (viewed as a 1×2 matrix) in terms of the 1×3 matrix ∇𝐹(𝑞) and a matrix of partial derivatives of 𝜙.

  • A ∇ 𝑤 ( 𝑝 ) = ∇ 𝐹 ( 𝑞 ) ⋅ ⎡ ⎢ ⎢ ⎢ ⎣ 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑥 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ∇ 𝑤 ( 𝑝 ) = ∇ 𝐹 ( 𝑞 ) ⋅ ⎡ ⎢ ⎢ ⎢ ⎣ 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑦 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ∇ 𝑤 ( 𝑝 ) = ∇ 𝐹 ( 𝑞 ) ⋅ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • D ∇ 𝑤 ( 𝑝 ) = ∇ 𝐹 ( 𝑞 ) ⋅ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
  • E ∇ 𝑤 ( 𝑝 ) = ∇ 𝐹 ( 𝑞 ) ⋅ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 𝜕 𝜙 𝜕 𝑥 𝜕 𝜙 𝜕 𝑦 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Q23:

Let 𝑤=𝐹(𝜙(𝑥,𝑦)) where 𝜙=𝑥+𝑦,𝑥−𝑦,𝑥𝑦 and 𝐹(𝑝,𝑞,𝑟)=10𝑝+6𝑞−16𝑟. Given that there is a line in the 𝑥-𝑦 plane for which ∇𝑤=0, find the equation of this line.

  • A 𝑦 = 𝑥
  • B 𝑦 = 2 𝑥
  • C 𝑦 = 𝑥 2
  • D 𝑦 = − 2 𝑥
  • E 𝑦 = − 𝑥

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