Glossary

3.3 Implicit differentiation

2 min read•august 6, 2024

is a powerful technique for finding derivatives of complex functions. It allows us to differentiate equations without isolating variables, making it especially useful for equations that are hard to solve explicitly.

This method builds on our understanding of and the . By applying these concepts to implicitly defined functions, we can tackle more advanced problems in multivariable calculus and real-world applications.

Implicit Differentiation

Differentiating Implicitly Defined Functions

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  • Implicit functions define relationships between variables without explicitly solving for one variable in terms of the others
  • Implicit differentiation differentiates both sides of an implicit equation with respect to an
  • Applies the chain rule to find the derivative of the with respect to the independent variable
  • Useful for finding the derivative of functions that are not easily solved for the dependent variable (yy in terms of xx)

Applying the Chain Rule and Total Differential

  • The chain rule for partial derivatives states that if z=f(x,y)z=f(x,y) and x=g(t)x=g(t) and y=h(t)y=h(t), then dzdt=∂f∂xdxdt+∂f∂ydydt\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}
  • The total differential of a function f(x,y)f(x,y) is df=∂f∂xdx+∂f∂ydydf = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy
  • Represents the infinitesimal change in ff resulting from infinitesimal changes in xx and yy
  • Useful for approximating small changes in a function and for implicit differentiation

Implicit Functions and Curves

Implicit Function Theorem and Level Curves

  • The states that if F(x,y)=0F(x,y)=0 and certain conditions are met, then there exists a unique function y=f(x)y=f(x) such that F(x,f(x))=0F(x,f(x))=0 in some neighborhood of a point (a,b)(a,b)
  • Guarantees the existence of an implicit function locally under certain conditions (∂F∂y≠0\frac{\partial F}{\partial y} \neq 0)
  • are curves in the xyxy-plane along which a function f(x,y)f(x,y) is constant
  • Defined by the equation f(x,y)=cf(x,y)=c for some constant cc
  • Examples of level curves include contour lines on a topographic map and iso-pressure curves on a weather map

Gradient Vector and Its Applications

  • The of a function f(x,y)f(x,y) is ∇f=(∂f∂x,∂f∂y)\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)
  • Points in the direction of the greatest rate of increase of ff at a given point
  • Perpendicular to the level curve of ff at any point
  • Useful for finding the direction of steepest ascent or descent (e.g., in optimization problems or in studying the flow of fluids or heat)

Key Terms to Review (21)

Chain Rule: The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if you have a function that is composed of other functions, you can find the derivative of the composite function by multiplying the derivative of the outer function by the derivative of the inner function. This rule plays a crucial role in calculating partial derivatives, implicit differentiation, and understanding how changes in one variable affect another through multi-variable functions.
Dependent Variable: A dependent variable is a variable in a mathematical equation or function that is affected by changes in one or more independent variables. This term highlights the relationship between variables, where the dependent variable's value is determined based on the values of independent variables. Understanding how dependent variables function is crucial in analyzing functions and equations, especially when using implicit differentiation to find rates of change.
Dy/dx: The term dy/dx represents the derivative of a function, indicating the rate of change of the dependent variable y with respect to the independent variable x. It is a fundamental concept in calculus that reveals how a function behaves at any given point, connecting directly to techniques like implicit differentiation and the chain rule. Understanding dy/dx allows for analyzing complex relationships in functions that may not be explicitly solvable.
Dynamics of Systems: Dynamics of systems refers to the study of how systems change and evolve over time, particularly through the influence of various factors or forces. This concept is crucial in understanding the relationships and interactions within systems, allowing for predictions about future behavior based on current conditions and external influences.
Euler Notation: Euler notation is a mathematical representation that uses complex exponentials to express trigonometric functions and is often employed in calculus and differential equations. This notation simplifies the manipulation of complex numbers and relates closely to Euler's formula, which states that for any real number $$x$$, $$e^{ix} = ext{cos}(x) + i ext{sin}(x)$$. By leveraging this relationship, Euler notation provides a powerful way to analyze oscillatory behavior in various mathematical contexts.
Function Definition: A function definition describes a specific relationship between inputs and outputs, where each input is associated with exactly one output. This concept is fundamental in calculus, especially when exploring the behavior of curves and finding slopes of tangent lines. Functions can be represented in various forms, including equations, graphs, and tables, which facilitate the understanding of how variables interact with one another.
Geometry of Curves: The geometry of curves focuses on the properties and characteristics of curves in a coordinate system, including their shapes, slopes, and how they behave under various transformations. Understanding the geometry of curves involves analyzing their curvature, concavity, and the relationships between tangent lines and the curves themselves, which are essential for applications like implicit differentiation.
Gradient Vector: The gradient vector is a vector that represents the direction and rate of the steepest ascent of a multivariable function. It is composed of partial derivatives and provides crucial information about how the function changes at a given point, linking concepts like optimization, directional derivatives, and surface analysis.
Higher-order derivatives: Higher-order derivatives are the derivatives of a function taken more than once. While the first derivative gives the rate of change or slope of the function, the second derivative reveals information about the curvature and concavity, and further derivatives can provide insights into the behavior of the function. They play a crucial role in understanding the dynamics of vector-valued functions, approximating functions through differentials, applying implicit differentiation, and utilizing the chain rule in complex functions.
Implicit Differentiation: Implicit differentiation is a technique used to differentiate equations that define a relationship between variables implicitly rather than explicitly. This method allows us to find the derivative of one variable in terms of another without solving for one variable in terms of the other, which is especially useful for complex functions or curves.
Implicit Function Theorem: The Implicit Function Theorem provides conditions under which a relation defined by an equation can be expressed as a function. Specifically, it states that if you have a function defined implicitly by an equation involving multiple variables, and if certain conditions are met (like the partial derivatives being non-zero), then you can locally solve for one variable in terms of others, effectively allowing us to treat the relation as a function. This concept connects deeply to how we analyze the domains and ranges of multivariable functions, differentiate implicitly, tackle constrained optimization problems, and understand the behavior of graphs and level curves.
Independent Variable: An independent variable is a variable in mathematical functions and experiments that is manipulated or controlled to test its effects on the dependent variable. In implicit differentiation, the independent variable often represents one of the variables in an equation, while its relationship with the dependent variable can be examined through differentiation techniques. Understanding the role of the independent variable is crucial for analyzing how changes in it can influence outcomes in equations and graphs.
Leibniz Notation: Leibniz notation is a formalism used in calculus to represent the derivative of a function. It is denoted as $$\frac{dy}{dx}$$, where $$y$$ is the dependent variable and $$x$$ is the independent variable. This notation emphasizes the relationship between the two variables and allows for clear differentiation in the context of implicit differentiation, where functions may not be explicitly solved for one variable in terms of another.
Level Curves: Level curves are the curves on a graph representing all points where a multivariable function has the same constant value. These curves provide insight into the behavior of functions with two variables by visually depicting how the output value changes with different combinations of input values, and they help to analyze critical points, gradients, and optimization problems.
Mean Value Theorem: The Mean Value Theorem states that for any continuous function that is differentiable on an open interval, there exists at least one point where the derivative equals the average rate of change of the function over that interval. This theorem bridges the gap between the function's behavior and its instantaneous rate of change, leading to important applications in approximation, implicit differentiation, and using the chain rule.
Normal Line: A normal line is a line that is perpendicular to a curve at a given point. It is essential in understanding the behavior of curves, especially when analyzing their slopes and the angles they form with tangent lines. The slope of the normal line can be found by taking the negative reciprocal of the slope of the tangent line at that same point, making it a key concept in implicit differentiation.
Partial Derivatives: Partial derivatives measure how a multivariable function changes as one variable changes while keeping other variables constant. This concept is crucial for understanding the behavior of functions with several variables and plays a significant role in various applications, such as optimization and the analysis of surfaces.
Polynomial functions: Polynomial functions are mathematical expressions that involve sums of powers of variables multiplied by coefficients. These functions can take various forms, including linear, quadratic, cubic, and higher-degree polynomials, and are defined by the general form $$f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$$ where the coefficients $$a_n, a_{n-1}, ..., a_0$$ are constants and $$n$$ is a non-negative integer. Understanding polynomial functions is crucial as they are used in approximations, differentiations, and applications in vector calculus.
Relation: A relation is a set of ordered pairs that connects elements from one set to another, often representing a relationship between two variables. In mathematics, relations can describe how inputs correspond to outputs, which is particularly important when dealing with functions or equations that are not explicitly solved for one variable in terms of another. Understanding relations allows us to analyze and interpret various mathematical situations, especially when differentiating implicitly.
Tangent Line: A tangent line is a straight line that touches a curve at a single point without crossing it at that point. This line represents the instantaneous rate of change of the curve at that point, showing how the function behaves in its immediate vicinity. The slope of the tangent line is equal to the derivative of the function at that specific point, which is especially relevant when working with implicit differentiation.
Trigonometric Functions: Trigonometric functions are mathematical functions that relate angles of a triangle to the ratios of its sides, commonly used in various fields of mathematics and applied sciences. These functions include sine, cosine, tangent, cosecant, secant, and cotangent, which help to model periodic phenomena and solve problems involving triangles and angles. Understanding these functions is essential as they are frequently utilized in approximations, implicit differentiation, and chain rule applications.
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