Examples using Implicit Differentiation

Implicit Differentiation
Implicit differentiation is a technique used in calculus to find the derivative of a function where the dependent variable (usually (y)) is not explicitly defined in terms of the independent variable (usually (x)).

In an explicit function, we have (y = f(x)). However, in an implicit function, the relationship between (x) and (y) is given by an equation of the form (F(x, y) = 0) or (F(x, y) = C), where (C) is a constant, and it might be difficult or impossible to solve for (y) explicitly.

Here are some examples to show how to use implicit differentiation.

The following diagrams show the steps for implicit differentiation. Scroll down the page for more examples and solutions on how to use implicit differentiation.


 

Steps to Perform Implicit Differentiation

1. Differentiate both sides of the equation with respect to x.
   Apply standard differentiation rules (power rule, product rule, quotient rule, chain rule, etc.) to every term on both sides of the equation.
   Whenever you differentiate a term involving y, multiply its derivative by \(\frac{dy}{dx}\).
   The derivative of any constant is 0.

2. Collect all terms containing \(\frac{dy}{dx}\) on one side of the equation (usually the left side) and move all other terms to the other side.

3. Factor out \(\frac{dy}{dx}\) from the terms collected on one side.

4. Solve for \(\frac{dy}{dx}\)by dividing both sides by the factored expression.

Videos

Implicit Differentiation
Examples:
Find dy/dx. Then find the slope of the tangent line at the given point.
a) 2x² - 3y³ = 5 at (-2,1)
b) y³ + x²y⁵ - x⁴ = 27 at (0,3)

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Implicit Differentiation - Basic Idea and Examples
Example:
Find dy/dx of x² + (xy) + cos(y) = 8y

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Implicit Differentiation Examples
An example of finding a tangent line is also given.
Example:

1. Find dy/dx of 1 + x = sin(xy²)
   
2. Find the equation of the tangent line at (1,1) on the curve x² + xy + y² = 3.

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More Implicit Differentiation Examples
Examples:

1. Find the dy/dx of x³ + y³ = (xy)
   
2. Find the dy/dx of (x²y) + (xy²) = 3x

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