Partial Derivatives Practice Problems

Partial derivatives are one of the most basic concepts in mathematics, especially multivariable calculus and are widely used in physics, engineering and economics among other fields. Partial derivatives are important for students as many topics such as gradient vectors, optimizations, and differential equations are based on it.

The scope of the following article is to give the reader a general idea of what partial derivatives are and review several exercises to solidify the concept.

What are Partial Derivatives?

Partial derivatives is a mathematical concept used in vector calculus and differential geometry. The term ‘partial’ indicates that if the function is dependent on more than one variable, then the derivative will be taken considering one variable to calculate the change concerning the chosen variable. The partial derivative of a function f(x, y) is represented by ∂f/∂x. Partial derivatives show how the value of a function changes as one of its variables changes while others stay the same.

Notation

• ∂f/∂x: Partial Derivative of f with respect to x
• f_x: Other common notation for the partial derivative of f with respect to x

Partial Derivatives Practice Problems

Example 1: Find the partial derivatives ∂f/∂x and ∂f/∂y for f(x, y) = 3x² y + 4y³.

Solution:

f(x, y) = 3x² y + 4y³

∂f/∂x = ∂/∂x (3x²y + 4y³)

= 3 . 2xy + 0 = 6xy

∂f/∂x = ∂/∂y (3x²y + 4y³)

= 3x² + 4 . 3y³ = 3x² + 12y²

Example 2: Given f(x,y) = ln(xy), compute ∂f/∂x and ∂f/∂y.

Solution:

f(x, y) = ln(xy)

Using the chain rule:

∂f/∂x = ∂/∂x ln(xy) =

1/xy . y

= y/xy = 1/x

∂f/∂y = ∂/∂y ln(xy)

= 1/xy . x

= x/xy = 1/y

Example 3: Determine the second order partial derivatives ∂²f/∂x², ∂f²/∂y², and ∂²f/∂x∂y for f(x, y) = x²e^y.

Solution:

f(x, y) = x²e^y

First, find the first order partial derivatives:

∂f/∂x = ∂/∂x(x²e^y) = 2xe^y

∂f/∂x = ∂/∂y(x²e^y) = x²e^y

Now, find second order partial derivatives:

∂²/∂x = ∂/∂x(2xe^y) = 2e^y

∂²/∂y² = ∂/∂y(x²e^y) = x²e^y

∂²f/∂x∂y = ∂/∂y(2xe^y) = 2xe^y

Example 4: For f(x, y) = x³ + y³ - 3xy, find all the first and second order partial derivatives.

Solution:

f(x, y) = x³ + y³ - 3xy

First order partial derivatives:

∂f/∂x = ∂/∂x(x³ + y³ - 3xy)

= 3x² - 3y

∂f/∂x = ∂/∂y(x³ + y³ − 3xy)

= 3y² - 3x

Second order partial derivatives:

∂²f/∂x² = ∂/∂x(3x² − 3y) = 6x

∂²f/∂y² = ∂/∂y(3y² - 3x) = 6y

∂²f/∂x∂y = ∂/∂y(3x² - 3y) = -3

∂²f/∂y/∂x = ∂/∂x(3y² - 3x) = -3

Example 5: Find the equation of the tangent plane to the surface z= x² + y²at the point (1, 1, 2).

Solution:

Function is f(x, y) = x² + y²

First, find the partial derivatives

∂f/∂x = 2x

∂f/∂x = 2y

At the point (1, 1, 2):

∂f/∂x (1, 1) = 2 ⋅ 1 = 2

∂f/∂x (1, 1) = 2 ⋅ 1 = 2

Equation of the tangent plane is:

z - z₀ = ∂f/∂x(x₀, y₀)(x - x₀) + ∂f/∂x(x₀, y₀)(y - y₀)

z − 2 = 2(x − 1) + 2(y − 1)

z − 2 = 2x − 2 + 2y − 2

z = 2x + 2y − 2

Example 6: Given f(x, y) = xe^xy, find ∂f/∂x and ∂f/∂y.

Solution:

Using the product rule:

f(x, y) = xe^xy

∂f/∂x = ∂/∂x(xe^xy)

= e^xy + x . e^xy . y

=e^xy . y = e^xy(1 + xy)

∂f/∂x = ∂/∂y(xe^xy)

= x . e^xy . x = x²e^xy

Example 7: For f(x, y) = sin(x + y), find ∂f/∂x and ∂f/∂y.

Solution:

f(x, y) = sin(x + y)

∂f/∂x = ∂/∂x sin(x + y)

= cos(x + y) . ∂/∂x(x + y)

= cos(x + y) . 1 = cos(x + y)

∂f/∂x = ∂/∂y sin(x + y)

= cos(x + y) . ∂/∂y(x + y)

= cos(x + y) . 1 = cos(x + y)

Example 8: Find the partial derivatives ∂f/∂x and ∂f/∂y f(x, y) = x² +xy + y².

Solution:

f(x, y) = x² + xy + y²

∂f/∂x = ∂/∂x(x² + xy + y)

= 2x + y

∂f/∂x = ∂/∂y(x² + xy + y²)

= x + 2y

Partial Derivatives Worksheet

Q1. Find the partial derivatives ∂f/∂x and ∂f/∂y for f(x, y) = 3x²y _ 4y².

Q2. Given f(x, y) = ln(x, y), compute ∂f/∂x and ∂f/∂y.

Q3. Determine the second order partial derivatives ∂²f/∂x², ∂²f/∂y², and ∂²f/∂x∂y for f(x, y) = x²e^y.

Q4. For f(x, y) = x³ + y3 −3xy, find all the first and second order partial derivatives.

Q5. Find all the 1st order derivatives of the given function f(u, v) = u²sin(u+v³) − sec(4u)tan⁻¹(2v

Q6. Find the equation of the tangent plane to the surface z = x² + y² at the point(1, 1, 2).

Q7. Locate and classify the critical points of f (x, y) = x² − 4xy + 4y².

Q8. Find all the 1st order derivatives of the given function f(x, y, z) = 4x³y² − e^zy⁴ + z³/ x² + 4y − x¹⁶ .

Q9. Letf(x, y)=(x-y)2. Determine the equations and shapes of the cross-sections whenx=0,y=0,x=y, and describe the level curves. Use a three-dimensional graphing tool to graph the surface.

Q10. If z = f(x, y) = x⁴y³ + 8x²y + y⁴ + 5x, then,

• ∂z/∂x =?
• ∂z/∂y =?

Conclusion

Partial derivatives are one of the core problems of the multivariable calculus theory and finding their realizations in different branches of knowledge. This way students will be able to appreciate it or the geometric and theoretical aspects behind it more and master the problems. In this article, the author gave an introduction, elaboration, and examples procedures of partial derivatives to help with understanding them.

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