Implicit differentiation – Advanced Examples

In the previous article, we have discussed the introduction part and some basic examples of Implicit differentiation. So in this article, we will discuss some advanced examples of implicit differentiation.

Implicit differentiation is a method that makes use of the chain rule to differentiate implicitly defined functions. It is generally not easy to find the function explicitly and then differentiate. Instead, we can differentiate f (x, y) and then solve the rest of the equation to find the value of . Even when it is possible to explicitly solve the original equation, the formula resulting from total differentiation is, in general, much simpler and easier to use.

Method to solve

• Differentiate both sides of the equation with respect to x.
• Follow the rules of differentiation.
• Use the chain rule to differentiate expressions involving y.
• Solve the equation for 

Implicit differentiation involves differentiating an implicit equation with respect to the desired variable x, while regarding the other variables as unspecified functions dependent on x.

To differentiate an implicit function, there are two common methods:

In the first method, the implicit equation is solved for ?,, expressing it explicitly in terms of x and then differentiating y with respect to x This approach is practical when y can be easily expressed in terms of x.

In the second method, y is considered as a function of x, and both sides of the implicit equation are differentiated with respect to x.

Example 1: Find the derivative of y = cos (5x – 3y)?

Solution:

Given equation:

y = cos(5x – 3y)

Step 1: Differentiating both sides wrt x,

Step 2: Using Chain Rule

Step 3: Expanding the above equation

 

Step 4: Taking all terms with dy/dx on LHS

Step 5: Taking dy/dx common from the LHS of equation

 

Step 6: Isolate dy/dx

 

Example 2: Find the derivative of (x² + y²) ³ = 5x²y²?

Solution:

Given equation:

(x² + y²)³ = 5x²y²

Differentiating both sides:

Example 3: Find the derivative of  ?

Solution:

Given equation:

Differentiating both sides:

Example 4: Find the derivative of y = ln(x)?

Solution:

Given equation:

y = ln(x)

=> e^y = x

Differentiating both sides:

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Implicit differentiation is a valuable technique used to differentiate implicitly defined functions with respect to a desired variable, typically x. It offers flexibility when explicit expressions for variables are difficult to obtain or when dealing with complex equations involving multiple variables. By treating one variable as a function of another and applying the chain rule, implicit differentiation enables the determination of derivatives even in cases where explicit differentiation is impractical.

This method is indispensable in various fields of mathematics, science, and engineering, providing a powerful tool for analyzing relationships between variables in implicit equations.

What is Implicit Differentiation?

Implicit differentiation is a technique used to find the derivative of a function that is defined implicitly by an equation, where one variable is not expressed explicitly in terms of the other.

When is Implicit Differentiation Used?

Implicit differentiation is used when it’s difficult or impossible to express one variable explicitly in terms of the other in a given equation, making it challenging to apply standard differentiation techniques.

How does Implicit Differentiation Work?

Implicit differentiation involves differentiating both sides of an implicitly defined equation with respect to the desired variable, treating the other variables as functions of that variable. The chain rule is applied to differentiate these functions.

What are Advantages of Implicit Differentiation?

Implicit differentiation allows us to find derivatives of functions defined implicitly without explicitly solving for one variable in terms of the other. This technique is particularly useful in cases where explicit differentiation is impractical or impossible.

Can any function be Differentiated Implicitly?

Implicit differentiation can be applied to any equation where one variable is implicitly defined in terms of another. However, the complexity of the resulting differentiation may vary depending on the equation’s structure.

What are Common Applications of Implicit dDifferentiation?

Implicit differentiation is commonly used in calculus, physics, engineering, and other fields to solve problems involving curves, surfaces, optimization, and related rates where relationships are defined implicitly.

Are there any Limitations or Challenges Associated with Implicit Differentiation?

Implicit differentiation may introduce additional complexity compared to explicit differentiation, and interpreting the resulting derivatives may require careful analysis, especially in cases involving multiple variables and complex equations.

What are Some Common Mistakes to Avoid when Using Implicit Differentiation?

Common mistakes include not applying the chain rule correctly, overlooking terms involving the derivative of y with respect to x, and missing terms when differentiating products or quotients within the equation.

Are there any Alternative Methods to Implicit Differentiation?

While implicit differentiation is a powerful technique, alternative methods such as parametric differentiation or explicit differentiation may be applicable depending on the specific problem and equation structure. However, implicit differentiation remains a crucial tool in many scenarios.