Implicit Differentiation
EXPLICIT: one variable is described in terms of one other variable.
y = 2x3 - 5
In other situations functions are described implicitly using the relationship between the two variables such as, x4 + y2 = 1 - 5xy.
IMPLICIT: variables are NOT is described in terms of one other variable.
x4 + y2 = 1 - 5xy
Functions described implicitly can be differentiated without being written explicitly. This type of differentiation is called Implicit Differentiation.
To differentiate implicitly both sides of the equation are differentiated with respect to x and the resulting equation solved for .
Let's look at some examples:
Step 1: Differentiate the x-terms on both side of the equation with respect to x. |
Apply Sum Rule.
Apply Power and Constant Rule.
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Step 2: Differentiate the y-terms. Factor into and take the derivative of normally resulting in the derivative times . |
Original Implicit Diff. Substitute |
Step 3: Solve for and simplify. |
Original
Subtract 2x to both sides:
Divide both sides by 2y.
Simplify
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Step 1: Differentiate the x-terms on both side of the equation with respect to x. |
Apply Difference Rule.
Apply Constant Multiple Rule.
Apply Power Rule.
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Step 2: Differentiate the y-terms. Factor into and take the derivative of normally resulting in the derivative times . |
Original
Implicitly differentiate using the power rule.
Implicitly differentiate using the product rule.
Substitute into the original equation:
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Step 3: Solve for and simplify. |
Original
Add 3x2 to both sides:
Subtract from both sides.
Factor out from the left-hand side:
Divide both sides by 9y2 - 5x.
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Step 1: Differentiate the x-terms on both side of the equation with respect to x. |
Apply Difference Rule.
Apply Constant Multiple Rule.
Apply Power Rule.
|
Step 2: Differentiate the y-terms. Factor into and take the derivative of normally resulting in the derivative times . |
Original
Implicitly differentiate ey using the natural exponent rule.
Implicitly differentiate using the power rule.
Substitute into the original equation:
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Step 3: Solve for and simplify. |
Original
Add to both sides:
Factor out from the left-hand side:
Divide both sides by ey + 1.
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Related Links: Math algebra General Differentiation Rules Exponential Differentiation Rules Algebra Topics |
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