Implicit Differentiation Techniques

Implicit Differentiation

• Differentiate both sides of an equation with respect to x, treating y as a function of x.
• The chain rule must be applied when differentiating terms involving y.
• Collect all terms involving dy/dx.
• Solve for dy/dx.

Example 5

• Verify a point (2, 4) lies on the curve x³ + y³ - 9xy = 0.
• Find the equation of the tangent line and normal line to the curve at that point.

Implicit Differentiation Problems

• A variety of implicit differentiation problems are presented, ranging from simple to complex, demonstrating various applications of the implicit differentiation technique.
• Examples include finding dy/dx for different equations.
• Problems involve finding the second derivative (d²y/dx²).

Critical Thinking Question

• Demonstrate three approaches to find dy/dx for implicit equations.
• Compare and contrast the varying methodologies.
• Highlight the importance of correct application of the differentiating techniques in implicit equations.

Graphing Functions – Extrema

• Identify relative extrema (maxima and minima) and absolute extrema on given intervals of functions.
• Find the intervals of increase and decrease by analyzing the first derivative.
• Relative maximums are points on a graph where the slope of the tangent changes from positive to negative.
• Relative minimums are points on a graph where the slope of the tangent changes from negative to positive.
• Absolute maximums are the highest points on the graph within the given interval.
• Absolute minimums are the lowest points on the graph within the given interval.

Absolute Extrema of Functions

• Defined on closed intervals [a, b], find the absolute extrema of functions.
• Critical points are identified using the first derivative, f'(x), to locate potential extrema or points where the slope is zero or undefined.
• Evaluate the function at the critical points and the endpoints of the interval.
• The largest and smallest values found are the absolute maximum and absolute minimum, respectively, within the given interval.

Explore the concept of implicit differentiation through various problems and examples. Apply the chain rule and learn how to find dy/dx while considering different methodologies. This quiz also includes critical thinking questions that encourage deeper understanding of the techniques involved.