Derivatives - Derivatives Of Implicit Functions

Derivatives - Derivatives of Implicit Functions

Slide 1:

Introduction to derivatives of implicit functions
Definition of an implicit function
Example: Equation of a circle and its implicit function

Slide 2:

Implicit differentiation and its steps
Example: Finding the derivative of an implicit function

Slide 3:

Differentiation rules for implicit functions
Example: Applying chain rule in implicit differentiation

Slide 4:

Derivatives of implicit functions involving trigonometric functions
Examples: Differentiating sin(x^2 + y^2) = xy and cos(xy^2) = x^2 + y^2

Slide 5:

Derivatives of implicit functions involving exponential and logarithmic functions
Examples: Differentiating e^x + ln(xy) = x^2 and ln(y) = xy

Slide 6:

Derivatives of implicit functions involving inverse trigonometric functions
Examples: Differentiating sin^(-1)(x + y) = x^2 + y^2 and tan^(-1)(x^2y) = x - y

Slide 7:

Derivatives of implicit functions involving hyperbolic functions
Examples: Differentiating sinh(x + y) = x^2 - y^2 and tanh(xy) = x^2 + y^2

Slide 8:

Derivatives of implicit functions involving logarithms of absolute values
Examples: Differentiating |x| + ln(|y|) = x^2 - y^2 and ln(|y|) = x^2 + y^2

Slide 9:

Implicit differentiation with multiple variables
Examples: Differentiating x^2 + 2xy + y^2 = 1 and x^2 + xy + y^2 = 4

Slide 10:

Applications of derivatives of implicit functions
Example: Finding tangent lines and normal lines to curves represented by implicit functions

Slide 11:

Practical examples of implicit functions in real-life scenarios
Example: Modeling population growth or decay using implicit functions

Slide 12:

Solving equations with implicit functions
Example: Finding the points of intersection between two curves represented by implicit functions

Slide 13:

Techniques for solving implicit differentiation problems
Example: Utilizing symmetry and known derivatives to simplify the process

Slide 14:

Common mistakes to avoid in implicit differentiation
Example: Misapplying the chain rule or differentiating constants incorrectly

Slide 15:

Applications of implicit differentiation in physics and engineering
Example: Finding rates of change in complex systems

Slide 16:

Comparison between implicit and explicit functions
Example: Contrasting the methods used to find derivatives for each type

Slide 17:

Limitations and challenges of implicit differentiation
Example: Dealing with multiple solutions or reducing equations to simpler forms

Slide 18:

Related rates problems involving implicit functions
Example: Finding the rate of change of one variable with respect to another in dynamic systems

Slide 19:

Techniques for handling higher-order derivatives of implicit functions
Example: Calculating second or higher derivatives for more complex equations

Slide 20:

Summary and key takeaways from the lecture on derivatives of implicit functions
Example: Emphasizing the importance of implicit differentiation in various fields

Slide 21:

Review of derivative rules for implicit functions
Example: Finding the derivative of an implicitly defined function using the power rule

Slide 22:

Implicit differentiation with respect to different variables
Examples: Differentiating with respect to x and y separately in equations such as x^2 + y^2 = 1

Slide 23:

Solving for higher derivatives of implicit functions
Example: Finding the second derivative of an implicitly defined function

Slide 24:

Implicit differentiation with logarithmic functions
Examples: Differentiating logarithmic functions such as ln(x^2 + y^2) = x - y

Slide 25:

Implicit differentiation with exponential functions
Examples: Differentiating exponential functions such as e^xy = x^2 + y^2

Slide 26:

Implicit differentiation with functions raised to a power
Examples: Differentiating functions with exponents such as (x + y)^2 = x^3 + y^4

Slide 27:

Implicit differentiation with inverse functions
Examples: Differentiating inverse functions such as arcsin(2x - y) = x + y

Slide 28:

Implicit differentiation with trigonometric functions
Examples: Differentiating trigonometric functions such as sin(xy) + cos(xy) = 1

Slide 29:

Implicit differentiation with hyperbolic functions
Examples: Differentiating hyperbolic functions such as sinh^2(x) - cosh^2(y) = 1

Slide 30:

Review exercises and practice problems for the derivatives of implicit functions
Example: Application-based problem involving a real-life scenario where implicit differentiation is used