About This Chapter
Partial Derivatives - Chapter Summary
Spend a little time with these video lessons to learn about the infinite points of change that exist when dealing with more than one variable in a function and why this scenario makes it tough to take a derivative at any given point. The video lessons in this chapter can help you understand how to take derivatives of complex functions one variable at a time.
Our expert instructors first define partial variables and show you how to represent partial derivatives of functions with respect to x and to y. They build on that base with a discussion on finding derivatives of second order and higher. The chapter closes with lessons on the chain rule and directional derivatives. The topics in this chapter cover:
- How to solve equations with partial derivatives
- Representing partial derivatives geometrically
- How to solve higher-order partial derivatives
- Obtaining tangent planes to a surface
- Using the chain rule with partial derivatives
- Working with directional derivatives, including gradient of f and min/max
Each of these videos explains the process of solving partial derivative equations using real-world examples in under 10 minutes each. Our instructors are all experienced math professionals and teachers and they present the lessons with entertaining graphics for a memorable visual experience. Use the lesson transcripts for follow-up study and take the self-assessments to measure your understanding of the concepts.
1. Solving Partial Derivative Equations
In this lesson, we'll introduce partial derivative equations and look into how they are solved. We'll explore several of the derivative rules and apply those rules to solve a few examples.
2. Higher-Order Partial Derivatives Definition & Examples
In this lesson, we define the partial derivative and then extend this concept to find higher-order partial derivatives. Examples are used to expand your knowledge and skill.
3. Tangent Plane to the Surface
In this lesson, we extend the idea of finding the equation of a tangent line to a very nice procedure for finding the equation of a plane tangent to a surface. Applications include the design of support surfaces for three-dimensional objects.
4. The Chain Rule for Partial Derivatives
When evaluating the derivative of composite functions of several variables, the chain rule for partial derivatives is often used. In this lesson, we use examples to explore this method.
5. Using the Chain Rule to Differentiate Complex Functions
If you've ever seen a complicated function, this lesson is for you. Most functions that we want to differentiate are complicated functions, for which no single derivative rule will work. In this lesson, learn how to use the chain rule to simplify nesting equations.
6. Directional Derivatives, Gradient of f and the Min-Max
Using partial derivatives and a direction vector, we can find the directional derivative. In this lesson, we clarify the meaning of the directional derivative and relate it to gradients, minimums, and maximums.
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- Limits of Sequences & Functions
- Calculating Derivatives
- Curve Sketching in Precalculus
- Differentiable Functions & Min-Max Problems
- Indefinite Integrals in Calculus
- Definite Integrals in Calculus
- Additional Topics in Calculus
- L'Hopital's Rule, Integrals & Series in Calculus
- Analytic Geometry in 3-Dimensions
- Calculus: Min/Max & Integrals
- Algebra: Differential Equations
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- Algebra: Number Theory & Abstract Algebra
- Additional Topics: Sets
- Additional Topics: Unions & Intersections
- Additional Topics: Graphing & Probability
- Additional Topics: Standard Deviation
- Additional Topics: Topology & Complex Variables
- Additional Topics: Trigonometry
- Additional Topics: Theorems, Analysis & Optimizing
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