Directional Derivatives — Practice Problems and Worked Solutions

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17 worked problems

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1. 1.

   Find the directional derivative in the indicated direction.

   $$f=x^2+y,P=\left(1,2\right),v=\left(1,0\right)$$

   Answer: $2$

   Step-by-step solution

   1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}1\\ 0\end{array}\right]$$
   2. $$D_{u}f=\nabla f(P)\cdot u=2$$
2. 2.

   Find the directional derivative in the indicated direction.

   $$f=xy,P=\left(2,3\right),v=\left(0,1\right)$$

   Answer: $2$

   Step-by-step solution

   1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}0\\ 1\end{array}\right]$$
   2. $$D_{u}f=\nabla f(P)\cdot u=2$$
3. 3.

   Find the directional derivative in the indicated direction.

   $$f=x^2+y^2,P=\left(1,-1\right),v=\left(3,4\right)$$

   Answer: $-\frac{2}{5}$

   Step-by-step solution

   1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}\frac{3}{5}\\ \frac{4}{5}\end{array}\right]$$
   2. $$D_{u}f=\nabla f(P)\cdot u=-\frac{2}{5}$$
4. 4.

   Find the directional derivative in the indicated direction.

   $$f=x^3-y,P=\left(2,0\right),v=\left(1,0\right)$$

   Answer: $12$

   Step-by-step solution

   1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}1\\ 0\end{array}\right]$$
   2. $$D_{u}f=\nabla f(P)\cdot u=12$$
5. 5.

   Find the directional derivative in the indicated direction.

   $$f=y^2+e^x,P=\left(0,1\right),v=\left(0,-1\right)$$

   Answer: $-2$

   Step-by-step solution

   1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}0\\ -1\end{array}\right]$$
   2. $$D_{u}f=\nabla f(P)\cdot u=-2$$
6. 6.

   Find the directional derivative in the indicated direction.

   $$f=x{y}^2,P=\left(1,2\right),v=\left(3,-4\right)$$

   Answer: $-\frac{4}{5}$

   Step-by-step solution

   1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}\frac{3}{5}\\ -\frac{4}{5}\end{array}\right]$$
   2. $$D_{u}f=\nabla f(P)\cdot u=-\frac{4}{5}$$
7. 7.

   Find the directional derivative in the indicated direction.

   $$f=x^2+y^2,P=\left(2,1\right),v=\left(1,0\right)$$

   Answer: $4$

   Step-by-step solution

   1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}1\\ 0\end{array}\right]$$
   2. $$D_{u}f=\nabla f(P)\cdot u=4$$
8. 8.

   Find the directional derivative in the indicated direction.

   $$f=xy,P=\left(1,1\right),v=\left(0,1\right)$$

   Answer: $1$

   Step-by-step solution

   1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}0\\ 1\end{array}\right]$$
   2. $$D_{u}f=\nabla f(P)\cdot u=1$$
9. 9.

   Find the directional derivative in the indicated direction.

   $$f=x^2-y,P=\left(3,2\right),v=\left(3,4\right)$$

   Answer: $\frac{14}{5}$

   Step-by-step solution

   1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}\frac{3}{5}\\ \frac{4}{5}\end{array}\right]$$
   2. $$D_{u}f=\nabla f(P)\cdot u=\frac{14}{5}$$
10. 10.

    Find the directional derivative in the indicated direction.

    $$f=x^3+y,P=\left(1,0\right),v=\left(1,0\right)$$

    Answer: $3$

    Step-by-step solution

    1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}1\\ 0\end{array}\right]$$
    2. $$D_{u}f=\nabla f(P)\cdot u=3$$
11. 11.

    Find the directional derivative in the indicated direction.

    $$f=xy+y,P=\left(2,3\right),v=\left(0,1\right)$$

    Answer: $3$

    Step-by-step solution

    1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}0\\ 1\end{array}\right]$$
    2. $$D_{u}f=\nabla f(P)\cdot u=3$$
12. 12.

    Find the directional derivative in the indicated direction.

    $$f=x^{2}y,P=\left(1,2\right),v=\left(3,-4\right)$$

    Answer: $\frac{8}{5}$

    Step-by-step solution

    1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}\frac{3}{5}\\ -\frac{4}{5}\end{array}\right]$$
    2. $$D_{u}f=\nabla f(P)\cdot u=\frac{8}{5}$$
13. 13.

    Find the directional derivative in the indicated direction.

    $$f=x+y^2,P=\left(0,2\right),v=\left(1,0\right)$$

    Answer: $1$

    Step-by-step solution

    1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}1\\ 0\end{array}\right]$$
    2. $$D_{u}f=\nabla f(P)\cdot u=1$$
14. 14.

    Find the directional derivative in the indicated direction.

    $$f=x^2+2y,P=\left(1,1\right),v=\left(4,3\right)$$

    Answer: $\frac{14}{5}$

    Step-by-step solution

    1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}\frac{4}{5}\\ \frac{3}{5}\end{array}\right]$$
    2. $$D_{u}f=\nabla f(P)\cdot u=\frac{14}{5}$$
15. 15.

    Find the directional derivative in the indicated direction.

    $$f=x+y^3,P=\left(2,1\right),v=\left(0,1\right)$$

    Answer: $3$

    Step-by-step solution

    1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}0\\ 1\end{array}\right]$$
    2. $$D_{u}f=\nabla f(P)\cdot u=3$$
16. 16.

    Find the directional derivative in the indicated direction.

    $$f=x{y}^2,P=\left(2,1\right),v=\left(1,0\right)$$

    Answer: $1$

    Step-by-step solution

    1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}1\\ 0\end{array}\right]$$
    2. $$D_{u}f=\nabla f(P)\cdot u=1$$
17. 17.

    Find the directional derivative in the indicated direction.

    $$f=x^2+y^2,P=\left(3,4\right),v=\left(3,4\right)$$

    Answer: $10$

    Step-by-step solution

    1. $$u=\frac{v}{\Vert v\Vert }=\left[\begin{array}{c}\frac{3}{5}\\ \frac{4}{5}\end{array}\right]$$
    2. $$D_{u}f=\nabla f(P)\cdot u=10$$

More from Multivariable Calculus

• Vectors in 3D
• Lines & Planes
• Partial Derivatives
• Gradients
• Tangent Planes
• Critical Points
• Double Integrals
• Polar Integrals
• Triple Integrals
• Divergence & Curl