How to Find the Partial Derivative of a Function (With Examples)
The concept of partial derivative plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Understanding partial derivatives helps students succeed in competitive exams and build a strong base for advanced topics in calculus, Physics, and Economics.
What Is Partial Derivative?
A partial derivative is defined as the rate at which a multivariable function changes as just one of its variables changes, while all other variables remain constant. The special symbol for partial derivative is ∂ (curly d). You’ll find this concept applied in multivariable calculus, optimization, and various physical science problems.
Key Formula for Partial Derivative
Here’s the standard formula: \( \frac{\partial f(x, y)}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h} \)
For functions with more variables, you keep all other variables constant except the one you are differentiating.
Cross-Disciplinary Usage
Partial derivatives are not only useful in Maths but also play an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in questions about gradient, heat and mass transfer, or economic optimization.
Step-by-Step Illustration
Example: Find the partial derivatives of \( f(x, y) = x^2y + \sin x + \cos y \).
2. Differentiate each term with respect to x:
• \( \sin x \rightarrow \cos x \)
• \( \cos y \) (constant with respect to x) → 0
3. So, \( \frac{\partial f}{\partial x} = 2xy + \cos x \)
4. Now, to find \( \frac{\partial f}{\partial y} \), treat x as constant:
5. Differentiate each term with respect to y:
• \( \sin x \) (constant with respect to y) → 0
• \( \cos y \rightarrow -\sin y \)
6. So, \( \frac{\partial f}{\partial y} = x^2 - \sin y \)
Speed Trick or Vedic Shortcut
Here’s a quick method to remember: When taking a partial derivative, always treat the variable you’re not differentiating as a constant. For example, if you’re asked for \( \frac{\partial}{\partial x}(2xy + 5y) \), simply treat y like a fixed number.
Trick: In partial derivatives, look for terms without the variable you are differentiating—those drop out to 0! This helps speed up calculations in competitive exams like JEE and Olympiads. More such shortcuts and insights are shared in Vedantu’s interactive live sessions.
Try These Yourself
- Find \( \frac{\partial}{\partial x} \) and \( \frac{\partial}{\partial y} \) for \( f(x, y) = 3x + 4y \).
- If \( f(x, y) = e^{xy} \), what is \( \frac{\partial f}{\partial x} \)?
- For \( f(x, y, z) = x^2 + yz \), find \( \frac{\partial f}{\partial y} \).
- Spot the difference: Is \( d/dx \) the same as \( \partial/\partial x \)?
Frequent Errors and Misunderstandings
- Confusing ordinary derivative (d/dx) with partial derivative (∂/∂x)
- Differentiating with respect to x but forgetting to treat y (or other variables) as constant
- Missing that terms without the variable you’re differentiating become zero
Comparison: Partial vs Ordinary Derivative
| Partial Derivative (∂/∂x) | Ordinary Derivative (d/dx) |
|---|---|
| Used for functions with 2 or more variables | Used for functions with only 1 variable |
| Keep all other variables constant except one | Change only the independent variable |
| Notation: ∂f/∂x | Notation: df/dx |
Relation to Other Concepts
The idea of partial derivative connects with differentiation and chain rule. It’s also the backbone for advanced topics like gradients, multivariable calculus, and double integrals.
Classroom Tip
A good way to remember partial derivatives: “When in doubt, freeze everything but one variable!” Teachers at Vedantu often use this “fridge rule” to simplify multivariable logistics for new learners.
Wrapping It All Up
We explored partial derivatives—from definition, formula, worked examples, mistakes, and links to other chapters. Practice different functions and use the tricks provided to improve problem-solving speed. For more detailed learning, explore step-by-step solutions and video sessions on Vedantu.
Explore Related Maths Topics
- Differentiation Formula – Rules and formulas for differentiating single and multivariable functions.
- Chain Rule – How to differentiate composite functions and apply chain rule to partial derivatives.
- Multivariable Calculus – Learn about calculus with more than one variable.
- Double Integral – Applications where partial derivatives play a role in finding areas and volumes.
- Gradient – The connection between gradients and vector calculus using partial derivatives.
FAQs on Partial Derivative – Concept, Formula & Practice
1. What is a partial derivative in simple terms?
A partial derivative measures how a function with multiple variables changes when we allow only one of its variables to change, while keeping all other variables constant. It is a fundamental concept in multivariable calculus used to analyse the rate of change of a function along a specific axis or direction. For example, for a function f(x, y), the partial derivative with respect to x, denoted as ∂f/∂x, tells us how f changes as we vary x, assuming y is a fixed value.
2. What is the fundamental formula for finding a partial derivative?
The fundamental formula for the partial derivative of a function f(x, y) with respect to x is based on the limit definition of a derivative. It is given by:
∂f/∂x = lim(h→0) [f(x+h, y) - f(x, y)] / h
Similarly, the formula with respect to y is:
∂f/∂y = lim(k→0) [f(x, y+k) - f(x, y)] / k
In practice, we use standard differentiation rules by treating other variables as constants.
3. How do you find the partial derivative of a multivariable function?
To find the partial derivative of a function with respect to one variable, follow these steps:
- Identify the variable you are differentiating with respect to (e.g., x).
- Treat all other variables in the function as constants (e.g., treat y, z as numbers).
- Apply the standard rules of single-variable differentiation (like the power rule, product rule, and chain rule) to the function.
- The resulting expression is the partial derivative with respect to the chosen variable.
4. What is the main difference between an ordinary derivative (d/dx) and a partial derivative (∂/∂x)?
The main difference lies in the type of function they are applied to.
- An ordinary derivative (d/dx) is used for functions of a single independent variable, like f(x). It measures the total rate of change of the function.
- A partial derivative (∂/∂x) is used for functions of two or more independent variables, like f(x, y). It measures the rate of change with respect to one variable while holding the others constant.
5. Why is it important to treat other variables as constants when finding a partial derivative?
Treating other variables as constants is the core principle that allows us to isolate the impact of a single variable on a multivariable function. Imagine a 3D surface representing a function z = f(x, y). Finding ∂z/∂x is like taking a vertical slice of that surface parallel to the x-axis and finding the slope of the curve formed by that slice. This simplifies a complex multi-dimensional problem into a series of manageable 2D slope calculations, helping us understand the function's behaviour in one specific direction at a time.
6. What are some key applications of partial derivatives in real-world scenarios?
Partial derivatives are crucial for modelling systems where outcomes depend on multiple factors. Key applications include:
- Economics: To find how a change in the price of one product affects the demand for another (cross-price elasticity).
- Physics: In thermodynamics (e.g., Maxwell's relations), electromagnetism (Maxwell's equations), and fluid dynamics to describe how quantities like pressure, temperature, and velocity change in space and time.
- Engineering: To analyse heat transfer, stress distribution in materials, and control systems.
- Machine Learning: To optimise model performance using algorithms like gradient descent, which relies on partial derivatives of a loss function.
7. How are partial derivatives used in optimisation problems?
Partial derivatives are fundamental to finding the maximum or minimum values of a multivariable function. To find these optimal points (known as critical points), we calculate the first-order partial derivatives with respect to each variable and set them all equal to zero. Solving this system of equations gives us the coordinates of potential maxima, minima, or saddle points. Further analysis using second-order partial derivatives (the Hessian matrix) helps classify these points.
8. What is a second-order partial derivative and what does it tell us?
A second-order partial derivative is obtained by differentiating a function twice. For a function f(x, y), there are four types: ∂²f/∂x², ∂²f/∂y², ∂²f/∂x∂y, and ∂²f/∂y∂x. They provide deeper insights into the function's geometry:
- ∂²f/∂x² and ∂²f/∂y² describe the concavity (how the surface curves) in the x and y directions, respectively.
- The mixed partial derivatives (∂²f/∂x∂y) describe how the slope in one direction changes as you move in another direction. They are essential for confirming if a critical point is a maximum, minimum, or saddle point.
9. What are some common mistakes students make when calculating partial derivatives?
Common mistakes often arise from confusing partial with ordinary differentiation. Key errors to avoid include:
- Forgetting to treat other variables as constants: Accidentally differentiating a variable like 'y' when finding ∂f/∂x.
- Incorrectly applying the product or chain rule: Forgetting that a variable treated as a constant might be part of a product, e.g., in ∂/∂x (x²y), 'y' is a constant multiplier.
- Dropping constant terms incorrectly: A term like 'y²' becomes 0 when differentiating with respect to x, but in a term like 'x²y', the 'y' remains.
- Confusing notation: Using 'd' when '∂' is required, which changes the meaning of the operation.
10. What does the symbol ∂ represent in mathematics?
The symbol ∂ is called the 'partial derivative sign' or simply 'partial'. It is also informally known as 'del', 'doh', or 'curly d'. Its purpose is to explicitly distinguish partial differentiation from ordinary (or total) differentiation, which uses the symbol 'd'. When you see ∂f/∂x, it is a clear instruction that 'f' is a function of multiple variables, and you should only differentiate with respect to 'x' while treating all other variables as fixed constants.
