Implicit Differentiation
The Organic Chemistry Tutor・2 minutes read
The video focuses on implicit differentiation problems, providing examples like finding dy/dx in equations involving x and y variables, with solutions such as -x/y or -3/4 at specific points. Differentiation rules are applied to calculate dy/dx in various equations, with the final expressions determined through implicit differentiation techniques.
Insights
- Implicit differentiation involves adding dy/dx when differentiating a y variable, resulting in -x/y as the solution for dy/dx in x^2 + y^2 = 100.
- The final expression for dy/dx in x^3 + 4xy + y^2 = 13, found using the product rule, is (-3x^2 - 4y) / (4x + 2y), with dy/dx evaluated at (1, 2) as -11/8.
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Recent questions
What is implicit differentiation?
Implicit differentiation involves differentiating equations with both x and y variables.
How do you find dy/dx using implicit differentiation?
To find dy/dx using implicit differentiation, differentiate both sides of the equation with respect to x.
What is the derivative of y^3 with respect to x?
The derivative of y^3 with respect to x is 3y^2 * dy/dx.
How is dy/dx calculated for the equation x^3 + 4xy + y^2 = 13?
Dy/dx for x^3 + 4xy + y^2 = 13 is found using the product rule.
How is the second derivative d^2y/dx^2 evaluated at (1, 2)?
The second derivative d^2y/dx^2 is calculated and then evaluated at the given point.
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Summary
00:00
Implicit Differentiation: Solving for dy/dx Problems
- Implicit differentiation problems are the focus of the video.
- Given the equation x^2 + y^2 = 100, the task is to find dy/dx.
- Differentiating y^3 with respect to x results in 3y^2 * dy/dx.
- Implicit differentiation involves adding dy/dx when differentiating a y variable.
- The solution for dy/dx in the equation is -x/y.
- Calculating the slope at the point (6, 8) yields dy/dx = -3/4.
- For the equation x^3 + 4xy + y^2 = 13, dy/dx is found using the product rule.
- The final expression for dy/dx is (-3x^2 - 4y) / (4x + 2y).
- At the point (1, 2), dy/dx is -11/8.
- Implicit differentiation is applied to the equation 5 - x^2 = sin(xy^2).
- The final expression for dy/dx is (-2x - y^2 * cos(xy^2)) / (2xy * cos(xy^2)).
- To find d^2y/dx^2 and evaluate it at (1, 2), the second derivative is calculated.
- The simplified second derivative at (1, 2) is -9/16.
