Partial Derivatives — Practice Problems and Worked Solutions

The formulas you need here

19 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Differentiate with respect to x, holding the other variables fixed.

   $$f=x^{2}y+3y,f_x=?$$

   Answer: $2xy$

   Step-by-step solution

   1. $$f_x=\frac{\partial }{\partial x}(x^{2}y+3y)$$
   2. $$=2xy$$
2. 2.

   Differentiate with respect to y, holding the other variables fixed.

   $$f=x{y}^2+\sin \left(x\right),f_y=?$$

   Answer: $2xy$

   Step-by-step solution

   1. $$f_y=\frac{\partial }{\partial y}(x{y}^2+\sin \left(x\right))$$
   2. $$=2xy$$
3. 3.

   Differentiate with respect to x, holding the other variables fixed.

   $$f=e^{xy},f_x=?$$

   Answer: $y{e}^{xy}$

   Step-by-step solution

   1. $$f_x=\frac{\partial }{\partial x}(e^{xy})$$
   2. $$=y{e}^{xy}$$
4. 4.

   Differentiate with respect to y, holding the other variables fixed.

   $$f=x^3-2xy+y^2,f_y=?$$

   Answer: $-2x+2y$

   Step-by-step solution

   1. $$f_y=\frac{\partial }{\partial y}(x^3-2xy+y^2)$$
   2. $$=-2x+2y$$
5. 5.

   Differentiate with respect to x, holding the other variables fixed.

   $$f=xy+\log \left(x\right),f_x=?$$

   Answer: $y+\frac{1}{x}$

   Step-by-step solution

   1. $$f_x=\frac{\partial }{\partial x}(xy+\log \left(x\right))$$
   2. $$=y+\frac{1}{x}$$
6. 6.

   Differentiate with respect to y, holding the other variables fixed.

   $$f=x^2{y}^2+xz,f_y=?$$

   Answer: $2x^{2}y$

   Step-by-step solution

   1. $$f_y=\frac{\partial }{\partial y}(x^2{y}^2+xz)$$
   2. $$=2x^{2}y$$
7. 7.

   Differentiate with respect to x, holding the other variables fixed.

   $$f=x^3+2y,f_x=?$$

   Answer: $3x^2$

   Step-by-step solution

   1. $$f_x=\frac{\partial }{\partial x}(x^3+2y)$$
   2. $$=3x^2$$
8. 8.

   Differentiate with respect to y, holding the other variables fixed.

   $$f=x{y}^2,f_y=?$$

   Answer: $2xy$

   Step-by-step solution

   1. $$f_y=\frac{\partial }{\partial y}(x{y}^2)$$
   2. $$=2xy$$
9. 9.

   Differentiate with respect to x, holding the other variables fixed.

   $$f=y^2+\cos \left(x\right),f_x=?$$

   Answer: $-\sin \left(x\right)$

   Step-by-step solution

   1. $$f_x=\frac{\partial }{\partial x}(y^2+\cos \left(x\right))$$
   2. $$=-\sin \left(x\right)$$
10. 10.

    Differentiate with respect to y, holding the other variables fixed.

    $$f=x^{2}z+y,f_y=?$$

    Answer: $1$

    Step-by-step solution

    1. $$f_y=\frac{\partial }{\partial y}(x^{2}z+y)$$
    2. $$=1$$
11. 11.

    Differentiate with respect to x, holding the other variables fixed.

    $$f=e^{xy},f_x=?$$

    Answer: $y{e}^{xy}$

    Step-by-step solution

    1. $$f_x=\frac{\partial }{\partial x}(e^{xy})$$
    2. $$=y{e}^{xy}$$
12. 12.

    Differentiate with respect to y, holding the other variables fixed.

    $$f=xy+yz,f_y=?$$

    Answer: $x+z$

    Step-by-step solution

    1. $$f_y=\frac{\partial }{\partial y}(xy+yz)$$
    2. $$=x+z$$
13. 13.

    Differentiate with respect to x, holding the other variables fixed.

    $$f=x^4-y^2,f_x=?$$

    Answer: $4x^3$

    Step-by-step solution

    1. $$f_x=\frac{\partial }{\partial x}(x^4-y^2)$$
    2. $$=4x^3$$
14. 14.

    Differentiate with respect to y, holding the other variables fixed.

    $$f=x^3+\sin \left(y\right),f_y=?$$

    Answer: $\cos \left(y\right)$

    Step-by-step solution

    1. $$f_y=\frac{\partial }{\partial y}(x^3+\sin \left(y\right))$$
    2. $$=\cos \left(y\right)$$
15. 15.

    Differentiate with respect to x, holding the other variables fixed.

    $$f=x^{2}y+z^2,f_x=?$$

    Answer: $2xy$

    Step-by-step solution

    1. $$f_x=\frac{\partial }{\partial x}(x^{2}y+z^2)$$
    2. $$=2xy$$
16. 16.

    Differentiate with respect to y, holding the other variables fixed.

    $$f=\frac{x}{y},f_y=?$$

    Answer: $-\frac{x}{y^2}$

    Step-by-step solution

    1. $$f_y=\frac{\partial }{\partial y}(\frac{x}{y})$$
    2. $$=-\frac{x}{y^2}$$
17. 17.

    Differentiate with respect to x, holding the other variables fixed.

    $$f=x^2+e^y,f_x=?$$

    Answer: $2x$

    Step-by-step solution

    1. $$f_x=\frac{\partial }{\partial x}(x^2+e^y)$$
    2. $$=2x$$
18. 18.

    Differentiate with respect to y, holding the other variables fixed.

    $$f=xyz,f_y=?$$

    Answer: $xz$

    Step-by-step solution

    1. $$f_y=\frac{\partial }{\partial y}(xyz)$$
    2. $$=xz$$
19. 19.

    Differentiate with respect to x, holding the other variables fixed.

    $$f=-xy+y^3,f_x=?$$

    Answer: $-y$

    Step-by-step solution

    1. $$f_x=\frac{\partial }{\partial x}(-xy+y^3)$$
    2. $$=-y$$

More from Multivariable Calculus

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