Determine D Y D X \frac{dy}{dx} D X D Y ​ For Each Equation:a. ( X + Y ) 3 = 12 X (x+y)^3 = 12x ( X + Y ) 3 = 12 X B. X + Y − 2 X = 1 \sqrt{x+y} - 2x = 1 X + Y ​ − 2 X = 1

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Introduction

Implicit differentiation is a powerful technique used in calculus to find the derivative of an implicitly defined function. In this article, we will explore how to use implicit differentiation to find the derivative of two given equations. We will start by understanding the concept of implicit differentiation and then apply it to each equation.

What is Implicit Differentiation?

Implicit differentiation is a method used to find the derivative of an implicitly defined function. An implicitly defined function is a function where the variable is not isolated on one side of the equation. In other words, the function is defined implicitly, and we need to use differentiation to find the derivative.

Step 1: Differentiate Both Sides of the Equation

The first step in implicit differentiation is to differentiate both sides of the equation with respect to the variable. This will give us an equation that involves the derivative of the function.

Step 2: Apply the Chain Rule

When differentiating an implicitly defined function, we need to apply the chain rule. The chain rule states that if we have a composite function of the form f(g(x)), then the derivative of the function is given by f'(g(x)) * g'(x).

Step 3: Solve for the Derivative

After applying the chain rule, we need to solve for the derivative of the function. This will give us the final answer.

Applying Implicit Differentiation to the First Equation

Let's apply implicit differentiation to the first equation: (x+y)3=12x(x+y)^3 = 12x.

Step 1: Differentiate Both Sides of the Equation

To differentiate both sides of the equation, we need to use the chain rule. The derivative of the left-hand side of the equation is given by:

ddx((x+y)3)=3(x+y)2ddx(x+y)\frac{d}{dx}((x+y)^3) = 3(x+y)^2 \frac{d}{dx}(x+y)

The derivative of the right-hand side of the equation is given by:

ddx(12x)=12\frac{d}{dx}(12x) = 12

Step 2: Apply the Chain Rule

Now, we need to apply the chain rule to the left-hand side of the equation. The derivative of (x+y)(x+y) is given by:

ddx(x+y)=ddx(x)+ddx(y)=1+dydx\frac{d}{dx}(x+y) = \frac{d}{dx}(x) + \frac{d}{dx}(y) = 1 + \frac{dy}{dx}

Substituting this into the equation, we get:

3(x+y)2(1+dydx)=123(x+y)^2 (1 + \frac{dy}{dx}) = 12

Step 3: Solve for the Derivative

Now, we need to solve for the derivative of the function. To do this, we can start by expanding the left-hand side of the equation:

3(x+y)2+3(x+y)2dydx=123(x+y)^2 + 3(x+y)^2 \frac{dy}{dx} = 12

Next, we can simplify the equation by combining like terms:

3(x+y)2+3(x+y)2dydx=123(x+y)^2 + 3(x+y)^2 \frac{dy}{dx} = 12

3(x+y)2(1+dydx)=123(x+y)^2 (1 + \frac{dy}{dx}) = 12

Now, we can divide both sides of the equation by 3(x+y)23(x+y)^2:

1+dydx=123(x+y)21 + \frac{dy}{dx} = \frac{12}{3(x+y)^2}

Subtracting 1 from both sides of the equation, we get:

dydx=123(x+y)2−1\frac{dy}{dx} = \frac{12}{3(x+y)^2} - 1

Simplifying the equation, we get:

dydx=4(x+y)2−1\frac{dy}{dx} = \frac{4}{(x+y)^2} - 1

Applying Implicit Differentiation to the Second Equation

Let's apply implicit differentiation to the second equation: x+y−2x=1\sqrt{x+y} - 2x = 1.

Step 1: Differentiate Both Sides of the Equation

To differentiate both sides of the equation, we need to use the chain rule. The derivative of the left-hand side of the equation is given by:

ddx(x+y)−ddx(2x)=12x+yddx(x+y)−2\frac{d}{dx}(\sqrt{x+y}) - \frac{d}{dx}(2x) = \frac{1}{2\sqrt{x+y}} \frac{d}{dx}(x+y) - 2

The derivative of the right-hand side of the equation is given by:

ddx(1)=0\frac{d}{dx}(1) = 0

Step 2: Apply the Chain Rule

Now, we need to apply the chain rule to the left-hand side of the equation. The derivative of (x+y)(x+y) is given by:

ddx(x+y)=ddx(x)+ddx(y)=1+dydx\frac{d}{dx}(x+y) = \frac{d}{dx}(x) + \frac{d}{dx}(y) = 1 + \frac{dy}{dx}

Substituting this into the equation, we get:

12x+y(1+dydx)−2=0\frac{1}{2\sqrt{x+y}} (1 + \frac{dy}{dx}) - 2 = 0

Step 3: Solve for the Derivative

Now, we need to solve for the derivative of the function. To do this, we can start by multiplying both sides of the equation by 2x+y2\sqrt{x+y}:

1+dydx−4x+y=01 + \frac{dy}{dx} - 4\sqrt{x+y} = 0

Next, we can simplify the equation by combining like terms:

dydx−4x+y=−1\frac{dy}{dx} - 4\sqrt{x+y} = -1

Now, we can add 4x+y4\sqrt{x+y} to both sides of the equation:

dydx=4x+y−1\frac{dy}{dx} = 4\sqrt{x+y} - 1

Conclusion

In this article, we have applied implicit differentiation to two given equations. We have used the chain rule to differentiate both sides of the equation and then solved for the derivative of the function. The final answers are:

  • For the first equation: dydx=4(x+y)2−1\frac{dy}{dx} = \frac{4}{(x+y)^2} - 1
  • For the second equation: dydx=4x+y−1\frac{dy}{dx} = 4\sqrt{x+y} - 1

Q&A: Implicit Differentiation

Q: What is implicit differentiation?

A: Implicit differentiation is a method used to find the derivative of an implicitly defined function. An implicitly defined function is a function where the variable is not isolated on one side of the equation.

Q: When should I use implicit differentiation?

A: You should use implicit differentiation when you are given an equation that involves a function and its derivative, and you need to find the derivative of the function.

Q: How do I apply implicit differentiation to an equation?

A: To apply implicit differentiation to an equation, you need to follow these steps:

  1. Differentiate both sides of the equation with respect to the variable.
  2. Apply the chain rule to the left-hand side of the equation.
  3. Solve for the derivative of the function.

Q: What is the chain rule?

A: The chain rule is a rule used to differentiate composite functions. It states that if we have a composite function of the form f(g(x)), then the derivative of the function is given by f'(g(x)) * g'(x).

Q: How do I apply the chain rule to an equation?

A: To apply the chain rule to an equation, you need to identify the outer and inner functions and then differentiate them separately. The derivative of the outer function is multiplied by the derivative of the inner function.

Q: What are some common mistakes to avoid when applying implicit differentiation?

A: Some common mistakes to avoid when applying implicit differentiation include:

  • Forgetting to apply the chain rule
  • Not differentiating both sides of the equation
  • Not solving for the derivative of the function
  • Making algebraic errors

Q: Can I use implicit differentiation to find the derivative of a function that is not implicitly defined?

A: No, you cannot use implicit differentiation to find the derivative of a function that is not implicitly defined. Implicit differentiation is only used to find the derivative of an implicitly defined function.

Q: How do I check my work when applying implicit differentiation?

A: To check your work when applying implicit differentiation, you need to:

  • Verify that you have differentiated both sides of the equation correctly
  • Verify that you have applied the chain rule correctly
  • Verify that you have solved for the derivative of the function correctly
  • Check your algebraic work for errors

Q: What are some real-world applications of implicit differentiation?

A: Some real-world applications of implicit differentiation include:

  • Finding the rate of change of a quantity that is related to another quantity
  • Modeling the behavior of a system that involves multiple variables
  • Solving optimization problems that involve multiple variables

Conclusion

In this article, we have provided a comprehensive guide to implicit differentiation, including a Q&A section that answers common questions about the topic. We hope that this article has provided you with a better understanding of implicit differentiation and how to apply it to different equations.