Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate → Calculus → Differential Calculus ↓


Implicit Differentiation


Implicit differentiation is a powerful technique in calculus that allows us to find the derivative of a function even when it is not easy or possible to solve the function for one variable in terms of the other. In simple terms, it helps us differentiate a function given in implicit form. An implicit function is an equation where the dependent and independent variables are intertwined and cannot be separated easily.

Understanding built-in functions

Implicit functions involve two or more variables that are related through an equation. For example, consider the equation of a circle:

x² + y² = r²

In this equation, x and y are variables that are connected in such a way that y cannot be easily solved for in terms of x or vice versa. This is in contrast to an explicit function, such as y = x², where y is expressed explicitly as a function of x.

Underlying differentiation process

Implicit differentiation involves a few main steps: differentiation, application of the chain rule, and solving for the desired derivative. Let's explore these steps with examples.

Step 1: Differentiation

Differentiate both sides of the underlying equation with respect to the independent variable. Often, this variable is x. Remember to apply the derivative rules you know, such as the multiplication rule and the power rule.

Step 2: Apply the chain rule

When differentiating terms involving a dependent variable, use the chain rule. For example, if differentiating y² with respect to x, think of y as a function of x (even if it can't be solved for x explicitly) and apply the chain rule:

d(y²)/dx = 2y * dy/dx

Step 3: Solve for the derivative

After differentiating, you will get an equation involving dy/dx. The goal is to solve this equation for dy/dx, which is the derivative of y with respect to x.

Examples of implicit differentiation

Example 1: Circle equation

Let us first differentiate the presented circle equation:

x² + y² = r²
  1. Differentiating both sides with respect to x gives:
    d(x²)/dx + d(y²)/dx = d(r²)/dx
  2. Uses of derivatives:
    2x + 2y(dy/dx) = 0
  3. Solution of dy/dx:
    2y(dy/dx) = -2x
    dy/dx = -x/y

Example 2: Ellipse equation

Consider the ellipse equation:

x²/a² + y²/b² = 1
  1. Differentiating both sides with respect to x gives:
    d(x²/a²)/dx + d(y²/b²)/dx = d(1)/dx
  2. Uses of derivatives:
    (2x/a²) + (2y/b²)(dy/dx) = 0
  3. Solution of dy/dx:
    (2y/b²)(dy/dx) = -2x/a²
    dy/dx = (-x/a²)/(y/b²) = -bx/ay

Looking at underlying differentiation

Implicit differentiation may be easier to understand by visual examples. Consider a simple circle centered at the origin:

Y X

This represents the implicit function x² + y² = r². If we want to find the slope of the tangent at any point on this circle, we use implicit differentiation. The result of dy/dx = -x/y tells us how the derivatives of x and y on the curve relate to each other.

Implicit differentiation in real-life problems

Physics applications

In physics, implicit differentiation can be useful when dealing with rates of change that cannot be easily separated. For example, consider a scenario where two objects are moving in such a way that their positions are related by an equation. Implicitly differentiating this gives the rate at which the positions change relative to each other.

Economics applications

In economics, implicit functions often arise in constraint optimization problems, such as Lagrange multipliers, where differentiation may be necessary to find equilibria or rates of change within constrained systems.

Exercises and practice problems

To master implicit differentiation, practice with different equations. Below are some exercises:

Exercise 1

Differentiate the following with respect to x:

xy = 7

Solution:

  1. Difference between the two sides:
    x(dy/dx) + y = 0
  2. Solution of dy/dx:
    x(dy/dx) = -y
    dy/dx = -y/x

Exercise 2

Find dy/dx for the following:

sin(xy) = x + y

Solution:

  1. Differentiate both sides using the chain rule:
    cos(xy)(x(dy/dx) + y) = 1 + dy/dx
  2. Rearrange to solve for dy/dx:
    cos(xy)x(dy/dx) + cos(xy)y = 1 + dy/dx
  3. Group the terms containing dy/dx:
    cos(xy)x(dy/dx) - dy/dx = 1 - cos(xy)y
  4. Factor out dy/dx:
    dy/dx(cos(xy)x - 1) = 1 - cos(xy)y
  5. Solve for dy/dx:
    dy/dx = (1 - cos(xy)y)/(cos(xy)x - 1)

Conclusion

Implicit differentiation is an important concept in calculus that enhances our ability to deal with complex relationships between variables. By following the steps of differentiation, correctly applying the chain rule, and solving for the desired derivative, we can tackle a wide range of mathematical and real-life problems. Practicing and applying these principles will build proficiency and intuition in dealing with implicit functions.


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