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
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 totally differentiate f(x, y) and then solve the rest of the equation to find the value of
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.
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Solve the equation for
Advanced Examples
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)
=> ey = x
Differentiating both sides: