Find dy/dx if y^x+x^y+x^x=a^b | x^x+x^y+y^x=a^b | #x^x+x^y+y^x=a^b | Differentiate y^x+x^y+x^x=a^b

Introduction

This tutorial will guide you through the process of finding the derivative dy/dx for the equation y^x + x^y + x^x = a^b. This is a common problem in calculus, particularly when dealing with implicit differentiation. Understanding how to differentiate equations involving both x and y is essential for students studying calculus, especially in Class 12.

Step 1: Understand the Equation

Before differentiating, it's important to clearly understand the equation:

• The equation is: [ y^x + x^y + x^x = a^b ]
• Here, y is a function of x (y = f(x)), and a^b is a constant.

Step 2: Differentiate Each Term

We will use implicit differentiation to find dy/dx. Differentiate each term with respect to x:

1. Differentiate (y^x):

   • Use the formula for differentiating functions in the form of (u^v): [ \frac{d}{dx}(y^x) = y^x \left(\ln(y) \cdot \frac{dx}{dx} + x \cdot \frac{dy}{dx} \cdot \frac{1}{y}\right) ]

2. Differentiate (x^y):

   • Apply the same formula: [ \frac{d}{dx}(x^y) = x^y \left(\frac{dy}{dx} \cdot \ln(x) + y \cdot \frac{1}{x}\right) ]

3. Differentiate (x^x):

   • This can be differentiated using the formula: [ \frac{d}{dx}(x^x) = x^x \left(\ln(x) + 1\right) ]

4. Differentiate the constant (a^b):

   • The derivative of a constant is 0.

Step 3: Set Up the Equation

Combine the derivatives from Step 2 into a single equation: [ y^x \left(\ln(y) + x \cdot \frac{dy}{dx} \cdot \frac{1}{y}\right) + x^y \left(\frac{dy}{dx} \cdot \ln(x) + y \cdot \frac{1}{x}\right) + x^x \left(\ln(x) + 1\right) = 0 ]

Step 4: Solve for dy/dx

Now, isolate dy/dx:

1. Rearrange the equation to group all terms with dy/dx on one side.
2. Factor out dy/dx: [ \text{(grouped terms)} \cdot \frac{dy}{dx} = -(\text{other terms}) ]
3. Solve for dy/dx: [ \frac{dy}{dx} = -\frac{\text{other terms}}{\text{(grouped terms)}} ]

Step 5: Substitute Back Values (if applicable)

If you have specific values for x and y, substitute them back into the equation to find a numerical value for dy/dx.

Conclusion

In this tutorial, you learned how to differentiate the equation y^x + x^y + x^x = a^b using implicit differentiation. The process involves differentiating each term, setting up a combined equation, and then solving for dy/dx. Practice these steps with different values to reinforce your understanding. As you continue your studies in calculus, mastering implicit differentiation will be invaluable for tackling complex equations.