Use Implicit Differentiation To Find $\frac{d Y}{d X}$ For $x \geq -4$.$\[ Y \sqrt{x+4} = X Y + 3 \\]$\[ \frac{d Y}{d X} = \ \square \\]

Introduction

Implicit differentiation is a technique used to find the derivative of an implicitly defined function. It is a powerful tool that allows us to find the derivative of a function even when it is not explicitly defined in terms of one variable. In this article, we will use implicit differentiation to find the derivative of the function $y \sqrt{x+4} = x y + 3$ for $x \geq -4$.

The Problem

The given function is $y \sqrt{x+4} = x y + 3$. We are asked to find the derivative of this function with respect to $x$ for $x \geq -4$. To do this, we will use implicit differentiation.

Implicit Differentiation

Implicit differentiation is a technique used to find the derivative of an implicitly defined function. It is based on the chain rule and the product rule of differentiation. The chain rule states that if we have a composite function of the form $f(g(x))$, then the derivative of this function is given by $f'(g(x)) \cdot g'(x)$. The product rule states that if we have a product of two functions of the form $f(x) \cdot g(x)$, then the derivative of this function is given by $f'(x) \cdot g(x) + f(x) \cdot g'(x)$.

To use implicit differentiation, we will differentiate both sides of the given equation with respect to $x$. We will then solve for $\frac{d y}{d x}$.

Differentiating Both Sides

We will start by differentiating both sides of the given equation with respect to $x$. We will use the chain rule and the product rule to do this.

Differentiating the Left Side

The left side of the equation is $y \sqrt{x+4}$. We will differentiate this with respect to $x$ using the product rule.

$$\frac{d}{dx} (y \sqrt{x+4}) = \frac{d y}{d x} \cdot \sqrt{x+4} + y \cdot \frac{d}{dx} (\sqrt{x+4})$$

We will now differentiate the square root term using the chain rule.

$$\frac{d}{dx} (\sqrt{x+4}) = \frac{1}{2} (x+4)^{-\frac{1}{2}} \cdot 1$$

Substituting this back into the previous equation, we get:

$$\frac{d}{dx} (y \sqrt{x+4}) = \frac{d y}{d x} \cdot \sqrt{x+4} + y \cdot \frac{1}{2} (x+4)^{-\frac{1}{2}}$$

Differentiating the Right Side

The right side of the equation is $x y + 3$. We will differentiate this with respect to $x$ using the product rule.

$$\frac{d}{dx} (x y + 3) = \frac{d y}{d x} \cdot x + y \cdot \frac{d}{dx} (x) + \frac{d}{dx} (3)$$

We will now differentiate the $x$ term using the power rule.

$$\frac{d}{dx} (x) = 1$$

Substituting this back into the previous equation, we get:

$$\frac{d}{dx} (x y + 3) = \frac{d y}{d x} \cdot x + y \cdot 1 + 0$$

Equating the Derivatives

We will now equate the derivatives of the left and right sides of the equation.

$$\frac{d}{dx} (y \sqrt{x+4}) = \frac{d}{dx} (x y + 3)$$

Substituting the expressions we found earlier, we get:

$$\frac{d y}{d x} \cdot \sqrt{x+4} + y \cdot \frac{1}{2} (x+4)^{-\frac{1}{2}} = \frac{d y}{d x} \cdot x + y \cdot 1$$

Solving for $\frac{d y}{d x}$

We will now solve for $\frac{d y}{d x}$.

$$\frac{d y}{d x} \cdot \sqrt{x+4} + y \cdot \frac{1}{2} (x+4)^{-\frac{1}{2}} = \frac{d y}{d x} \cdot x + y \cdot 1$$

We will start by moving all the terms involving $\frac{d y}{d x}$ to one side of the equation.

$$\frac{d y}{d x} \cdot \sqrt{x+4} - \frac{d y}{d x} \cdot x = y \cdot 1 - y \cdot \frac{1}{2} (x+4)^{-\frac{1}{2}}$$

We will now factor out $\frac{d y}{d x}$ from the left side of the equation.

$$\frac{d y}{d x} \cdot (\sqrt{x+4} - x) = y \cdot (1 - \frac{1}{2} (x+4)^{-\frac{1}{2}})$$

We will now divide both sides of the equation by $(\sqrt{x+4} - x)$.

$$\frac{d y}{d x} = \frac{y \cdot (1 - \frac{1}{2} (x+4)^{-\frac{1}{2}})}{\sqrt{x+4} - x}$$

Simplifying the Expression

We will now simplify the expression for $\frac{d y}{d x}$.

$$\frac{d y}{d x} = \frac{y \cdot (1 - \frac{1}{2} (x+4)^{-\frac{1}{2}})}{\sqrt{x+4} - x}$$

We will start by multiplying the numerator and denominator by $2$ to eliminate the fraction in the numerator.

$$\frac{d y}{d x} = \frac{2 y \cdot (1 - \frac{1}{2} (x+4)^{-\frac{1}{2}})}{2 (\sqrt{x+4} - x)}$$

We will now simplify the numerator.

$$\frac{d y}{d x} = \frac{2 y - y (x+4)^{-\frac{1}{2}}}{2 (\sqrt{x+4} - x)}$$

We will now simplify the denominator.

$$\frac{d y}{d x} = \frac{2 y - y (x+4)^{-\frac{1}{2}}}{2 \sqrt{x+4} - 2 x}$$

We will now simplify the expression further.

$$\frac{d y}{d x} = \frac{y (2 - (x+4)^{-\frac{1}{2}})}{\sqrt{x+4} - x}$$

Final Answer

The final answer is $\boxed{\frac{y (2 - (x+4)^{-\frac{1}{2}})}{\sqrt{x+4} - x}}$.

Conclusion

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Introduction

Implicit differentiation is a technique used to find the derivative of an implicitly defined function. It is a powerful tool that allows us to find the derivative of a function even when it is not explicitly defined in terms of one variable. In this article, we will use implicit differentiation to find the derivative of the function $y \sqrt{x+4} = x y + 3$ for $x \geq -4$. We will also provide a Q&A section to help clarify any doubts.

Q&A

Q: What is implicit differentiation?

A: Implicit differentiation is a technique used to find the derivative of an implicitly defined function. It is based on the chain rule and the product rule of differentiation.

Q: How do I use implicit differentiation?

A: To use implicit differentiation, you need to differentiate both sides of the equation with respect to $x$. You will then solve for $\frac{d y}{d x}$.

Q: What is the chain rule?

A: The chain rule is a rule of differentiation that states that if we have a composite function of the form $f(g(x))$, then the derivative of this function is given by $f'(g(x)) \cdot g'(x)$.

Q: What is the product rule?

A: The product rule is a rule of differentiation that states that if we have a product of two functions of the form $f(x) \cdot g(x)$, then the derivative of this function is given by $f'(x) \cdot g(x) + f(x) \cdot g'(x)$.

Q: How do I differentiate a square root term?

A: To differentiate a square root term, you need to use the chain rule. The derivative of $\sqrt{x}$ is $\frac{1}{2} x^{-\frac{1}{2}}$.

Q: How do I simplify an expression involving a square root?

A: To simplify an expression involving a square root, you need to rationalize the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator.

Q: What is the final answer for the derivative of $y \sqrt{x+4} = x y + 3$?

A: The final answer is $\boxed{\frac{y (2 - (x+4)^{-\frac{1}{2}})}{\sqrt{x+4} - x}}$.

Q: Can I use implicit differentiation to find the derivative of any function?

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

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

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

• Forgetting to differentiate both sides of the equation
• Forgetting to solve for $\frac{d y}{d x}$
• Not rationalizing the denominator
• Not simplifying the expression

Conclusion

In this article, we used implicit differentiation to find the derivative of the function $y \sqrt{x+4} = x y + 3$ for $x \geq -4$. We also provided a Q&A section to help clarify any doubts. We hope this article has been helpful in understanding implicit differentiation.

Additional Resources

• Implicit Differentiation Tutorial
• Implicit Differentiation Examples
• Implicit Differentiation Practice Problems

Final Thoughts

Implicit differentiation is a powerful tool that allows us to find the derivative of an implicitly defined function. It is based on the chain rule and the product rule of differentiation. By following the steps outlined in this article, you can use implicit differentiation to find the derivative of any implicitly defined function.