Partial Derivatives in Engineering Mathematics

Partial derivatives are a basic concept in multivariable calculus. They convey how a function would change when one of its input variables changes, while keeping all the others constant. This turns out to be particularly useful in fields such as physics, engineering, economics, and computer science, where often systems depend on more than one variable.

What are Partial Derivatives?

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with all other variables held constant. For a function f(x,y), the partial derivative with respect to x, denoted as ∂f/∂x, measures the rate at which f changes as x changes, while y remains fixed.

Notation and Calculation

The most common notation for partial derivatives includes ∂f/∂x and f_x for the partial derivative of f with respect to x.

To calculate a partial derivative:

1. Identify the variable with respect to which you are differentiating.
2. Treat all other variables as constants.
3. Differentiate the function with respect to the chosen variable using the rules of differentiation.

Example: For the function f(x,y)=x2y+3xy2:

The partial derivative with respect to x is:

∂f/∂x =2xy+3y2

The partial derivative with respect to y is:

∂f/∂y =x2 +6xy

Properties of Partial Derivatives

Higher-Order Partial Derivatives

Partial derivatives of a given function of higher order are obtained when the function is differentiated successively with respect to one or more variables. If there are two independent variables, say x and y, in a function, f(x,y), then the second order partial derivatives are:

• ∂2f/∂x² : The second partial derivative with respect to x.
• ∂²f/∂y² : The second partial derivative with respect to y.
• ∂²f/∂x∂y or ∂²f/∂y∂x: The mixed partial derivatives.

Example: For f(x,y)=sin(xy), the second-order partial derivatives are:

• ∂2f/∂x2 = −y²sin(xy)
• ∂2f/∂y2 = −x²sin(xy)
• ∂2f/∂x∂y or ∂2f/∂y∂x = cos(xy)−xysin(xy)

Computing Partial Derivatives

Computing partial derivatives involves systematic application of differentiation rules:

• Identify the variable of interest: Determine which variable you are differentiating with respect to.
• Apply differentiation rules: Use standard rules (product, chain, quotient) while treating all other variables as constants.
• Check for higher-order derivatives: If needed, differentiate again to obtain higher-order derivatives.

Example: For the function f(x,y,z)=e^xy ⋅z³ :

• ∂f/∂x = y⋅e^xy⋅z³
• ∂f/∂y = x.e^xy ⋅z³
• ∂f/∂z = 3z² .e^xy

Partial Derivatives in Engineering Mathematics: A function is like a machine that takes some input and gives a single output. For example, y = f(x) is a function in 'x'. Here, we say 'x' is the independent variable and 'y' is the dependent variable as the value of 'y' depends on 'x'. 

Some examples of functions are:

1. f(x) = x² + 3 is an algebraic function.
2. e^x is the exponential function.
3. sin(x), cos(x), tan(x),...etc. are all trigonometric functions.

Now, all these functions are functions of a single variable, i.e. there is only one independent variable.

Partial Derivatives in Engineering Mathematics

Partial derivatives are a fundamental concept in multivariable calculus, often used in engineering mathematics to analyze how functions change when varying one variable while keeping others constant. This is crucial in fluid dynamics, thermodynamics, and structural analysis.

To understand the concept of partial derivative, we must first look at what a function in two variables means. 
Consider a function of the form z = f(x,y) where 'x' and 'y' are the independent variables and 'z' is the dependent variable. This function is called a function in two variables. Similarly, functions of several variables (i.e. with more than 2 independent variables) can also be defined. 

Partial Derivatives Examples

Some examples of multivariable functions or functions of several variables are:

1. f(x,y) = x² +y

2. f(x,y,z) = x-3y+4z

Let us visualize this concept through graph. First we consider a single variable function f(x) = x². 

Unlike functions of a single variable, we cannot visualize multivariable functions as a 2-D graph. For this, we plot it on the 3-D plane. For example, consider the graph of f(x,y) = x²+y²

For functions of several variables we define the limit as follows:

This means, finding the limit of f(x) as 'x' approaches 'a' and 'y' approaches 'b'. 

Similarly, the definitions of continuity and differentiability can be extended from definitions for single variable functions. 

Recall that the derivative of a function of single variable y=f(x) is defined as : f'(x) = `\frac{dy}{dx} 

For a function z = f(x,y) of two variables, we define the derivative as : \frac{\partial z}{\partial x}

This means calculating the derivative of function 'z' with respect to 'x' by keeping 'y' constant. Similarly, we can calculate the derivative of 'z' with respect to 'y' by keeping 'x' as constant as \frac{\partial z}{\partial y}

Geometrical Interpretation of Partial Derivative

As we know, for single variable functions, the derivative is computed as the slope of the tangent passing through the curve. Similarly, we can understand the geometric interpretation of a partial derivative of a multivariable function.

Consider a function of two variables, z = f(x,y) on the 3-D plane and let a plane y=b pass through the curve f(x,y). 

Now, we draw another curve f(x,b) lying on z that is perpendicular to the plane y=b. Consider two arbitrary points P,R on this curve and draw the secant passing through these points. 

The slope of this secant is calculated using the first principles as follows :

m = \frac{Δz}{Δx} = \frac{f(x+Δx,b)-f(a,b)}{Δx}

As the two points move closer to each other, the difference Δx approaches 0 and we calculate this in the form of the limit : \lim_{Δx\to0} \frac{Δz}{Δx} = \frac{f(a+Δx,b)-f(a,b)}{Δx}

This limit is the partial derivative of 'z' with respect to 'x' by treating 'y' as constant i.e. 

\frac{\partial z}{\partial x} = \frac{f(a+Δx,b)-f(a,b)}{Δx}

Calculation of Partial Derivatives of a Function

Steps to calculate partial derivative of a given function :

1. Consider z = f(x,y).
2. Compute partial derivative with respect to 'x' i.e. \frac{\partial z}{\partial x}     by considering 'y' as constant and differentiate the function with respect to 'x'.
3. Compute partial derivative with respect to 'y' i.e. \frac{\partial z}{\partial y}     by considering 'x' as constant and differentiate the function with respect to 'y'.

Example : z = x^2 + y^2 + 3xy

Here, for the given function, we calculate the two partial derivatives as follows : 

Case 1: Differentiating with respect to 'x' by treating 'y' as constant i.e. \frac{\partial z }{\partial x}

Case 2: Differentiating with respect to 'y' by treating 'x' as constant i.e. \frac{\partial z }{\partial y}

Second-Order Partial Derivatives

Similar to the computation of second-order derivatives for functions of single variables, we can compute the same for functions of several variables.

For an example we consider the same function z = x^2 + y^2 + 3xy   .

Case 1: We differentiate \frac{\partial z}{\partial x}     again with respect to 'x'

Case 2: We differentiate \frac{\partial z}{\partial y}     again with respect to 'y'

Case 3: We differentiate \frac{\partial z}{\partial x}      again with respect to 'y'

Case 4: We differentiate \frac{\partial z}{\partial y}      again with respect to 'x'

People Also Read:

• Partial Derivative: Definition, Formulas and Examples
• Application of Partial Derivative

Applications of Partial Derivatives in Engineering

Partial derivatives are widely used in various engineering disciplines to solve problems involving multiple variables:

• Heat Transfer: Describing the change in temperature distribution over time and space.
• Fluid Dynamics: Analyzing velocity fields and pressure distributions in fluid flows.
• Structural Analysis: Determining stress and strain in materials under load.

Solved Examples

Basic partial differentiation:

Given f(x,y) = x^2y + 3xy^2, find ∂f/∂x and ∂f/∂y.

Solution:

∂f/∂x = 2xy + 3y^2 (treat y as a constant)

∂f/∂y = x^2 + 6xy (treat x as a constant)

Higher-order partial derivatives:

For f(x,y) = x^3y^2 + 2xy, find ∂²f/∂x², ∂²f/∂y², and ∂²f/∂x∂y.

Solution:

∂f/∂x = 3x^2y^2 + 2y

∂²f/∂x² = 6xy^2

∂f/∂y = 2x^3y + 2x

∂²f/∂y² = 2x^3

∂f/∂x = 3x^2y^2 + 2y

∂²f/∂x∂y = 6xy^2 + 2

Chain rule for partial derivatives:

If z = f(x,y) where x = r cos θ and y = r sin θ, express ∂z/∂r and ∂z/∂θ in terms of ∂z/∂x and ∂z/∂y.

Solution:

∂z/∂r = (∂z/∂x)(∂x/∂r) + (∂z/∂y)(∂y/∂r)

= (∂z/∂x)(cos θ) + (∂z/∂y)(sin θ)

∂z/∂θ = (∂z/∂x)(∂x/∂θ) + (∂z/∂y)(∂y/∂θ)

= (∂z/∂x)(-r sin θ) + (∂z/∂y)(r cos θ)

Implicit differentiation:

Given x^2 + y^2 + z^2 = 1, find ∂z/∂x and ∂z/∂y.

Solution:

Differentiate with respect to x:

2x + 2y(∂y/∂x) + 2z(∂z/∂x) = 0

∂z/∂x = -x/z

Differentiate with respect to y:

2y + 2x(∂x/∂y) + 2z(∂z/∂y) = 0

∂z/∂y = -y/z

Gradient:

Find the gradient of f(x,y,z) = 2x^2y + yz^3 - 3xz.

Solution:

∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z)

= (4xy - 3z, 2x^2 + z^3, 3yz^2 - 3x)

Directional derivative:

For f(x,y) = x^2 + 2xy + y^2, find the directional derivative at (1,2) in the direction of u = (3/5, 4/5).

Solution:

∇f = (2x + 2y, 2x + 2y)

At (1,2): ∇f = (6, 6)

Directional derivative = ∇f · u

= (6, 6) · (3/5, 4/5)

= (6 * 3/5) + (6 * 4/5)

= 18/5 + 24/5

= 42/5 = 8.4

Partial differential equation:

Verify that u(x,t) = e^(-at) sin(x) is a solution to the heat equation ∂u/∂t = k(∂²u/∂x²).

Solution:

∂u/∂t = -ae^(-at) sin(x)

∂u/∂x = e^(-at) cos(x)

∂²u/∂x² = -e^(-at) sin(x)

Substituting into the heat equation:

-ae^(-at) sin(x) = k(-e^(-at) sin(x))

This is true if a = k, verifying the solution.

Laplacian:

Find the Laplacian of f(x,y,z) = x^2y + yz^2 + xz.

Solution:

∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²

∂²f/∂x² = 2y

∂²f/∂y² = 0

∂²f/∂z² = 2y

∇²f = 2y + 0 + 2y = 4y

Summary

Partial derivatives are a fundamental concept in engineering mathematics, allowing us to analyze how a function changes with respect to one variable while holding others constant. They are crucial in various engineering applications, including optimization, thermodynamics, and fluid dynamics. The process involves treating all variables except the one of interest as constants and then differentiating normally. Key rules include the product rule, chain rule, and implicit differentiation. Practice with diverse functions, including exponentials, logarithms, and trigonometric functions, is essential for mastery. Understanding higher-order and mixed partial derivatives further extends the utility of this concept in engineering analysis and problem-solving.

Related Articles:

• Application of Partial Derivatives in Engineering Mathematics
• Engineering Mathematics – Well Formed Formulas (WFF)