Implicit differentiation is a calculus technique used to find the derivative of functions where one variable is not explicitly expressed in terms of another. It's essential for equations like circles, where y is not isolated, and for calculating slopes of tangents and normals to curves at any point. The process involves differentiating both sides of an equation with respect to x and applying rules like the chain and product rules. Higher order derivatives can also be computed using this method.
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Implicit differentiation is a fundamental technique in calculus for finding the derivative of a function
Implicit functions are equations where y is not isolated on one side of the equation
The chain rule is used when differentiating terms involving y in implicit differentiation
In implicit differentiation, each term of the equation is differentiated with respect to x
The product rule is used when differentiating terms involving products of x and y
Implicit differentiation can be extended to compute higher order derivatives of implicit functions
Implicit differentiation is useful for determining the equations of tangent lines to curves at specific points
The slope of the normal line is the negative reciprocal of the slope of the tangent line, and can be found using implicit differentiation
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Implicit Function Example
Equation x^2 + y^2 = r^2 represents a circle, not solved for y explicitly.
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Differentiation Technique
Derive both sides with respect to x, apply chain rule for y terms, include dy/dx.
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Tangent Line Slope Calculation
Use implicit differentiation to find slope of tangent without solving y explicitly.
