Use Implicit Differentiation To Find D Y D X \frac{dy}{dx} D X D Y ​ . 39 Tan ⁡ − 1 Y 2 − Π X 3 Y = 4 Π 39 \tan^{-1} Y^2 - \pi X^3 Y = 4 \pi 39 Tan − 1 Y 2 − Π X 3 Y = 4 Π (Use Symbolic Notation And Fractions Where Needed.) D Y D X = □ \frac{dy}{dx} = \square D X D Y ​ = □

Introduction

Implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. It is a method that allows us to differentiate functions that are not explicitly defined in terms of a single variable. In this article, we will use implicit differentiation to find the derivative of the function $39 \tan^{-1} y^2 - \pi x^3 y = 4 \pi$.

What is 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 main idea behind implicit differentiation is to differentiate both sides of an equation with respect to a variable, while treating the other variable as a constant.

Step 1: Differentiate Both Sides of the Equation

To find the derivative of the function $39 \tan^{-1} y^2 - \pi x^3 y = 4 \pi$, we need to differentiate both sides of the equation with respect to $x$. We will use the chain rule and the product rule of differentiation to do this.

Differentiating the Left-Hand Side

The left-hand side of the equation is $39 \tan^{-1} y^2 - \pi x^3 y$. We can differentiate this using the chain rule and the product rule.

import sympy as sp
x, y = sp.symbols('x y')

lhs = 39sp.atan(y**2) - sp.pix**3*y

d_lhs_dx = sp.diff(lhs, x)

Differentiating the Right-Hand Side

The right-hand side of the equation is $4 \pi$. We can differentiate this using the power rule of differentiation.

# Define the right-hand side of the equation
rhs = 4*sp.pi

d_rhs_dx = sp.diff(rhs, x)

Step 2: Equate the Derivatives

Now that we have differentiated both sides of the equation, we can equate the derivatives.

# Equate the derivatives
equation = sp.Eq(d_lhs_dx, d_rhs_dx)

Step 3: Solve for $\frac{dy}{dx}$

Finally, we can solve for $\frac{dy}{dx}$ by isolating it on one side of the equation.

# Solve for dy/dx
dy_dx = sp.solve(equation, y)[0]

The Final Answer

The final answer is $\boxed{\frac{3 \pi x^2 y + 39 \frac{2y}{1+y^4}}{39 \frac{2y}{1+y^4} - 3 \pi x^2 y}}$.

Conclusion

Implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. In this article, we used implicit differentiation to find the derivative of the function $39 \tan^{-1} y^2 - \pi x^3 y = 4 \pi$. We differentiated both sides of the equation, equated the derivatives, and solved for $\frac{dy}{dx}$. The final answer is $\boxed{\frac{3 \pi x^2 y + 39 \frac{2y}{1+y^4}}{39 \frac{2y}{1+y^4} - 3 \pi x^2 y}}$.

Example Use Cases

Implicit differentiation has many practical applications in mathematics and science. Here are a few example use cases:

• Finding the derivative of an implicitly defined function
• Solving optimization problems
• Finding the equation of a tangent line to a curve
• Finding the equation of a normal line to a curve

Tips and Tricks

Here are a few tips and tricks to keep in mind when using implicit differentiation:

• Make sure to differentiate both sides of the equation with respect to the same variable.
• Use the chain rule and the product rule of differentiation as needed.
• Equate the derivatives and solve for the desired variable.
• Check your work by plugging in the solution back into the original equation.

Conclusion

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Introduction

Implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. In this article, we will answer some common questions about implicit differentiation.

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: When should I use implicit differentiation?

A: You should use implicit differentiation when you are given an implicitly defined function and you need to find its derivative. This is often the case in optimization problems, where you need to find the maximum or minimum of a function.

Q: How do I use implicit differentiation?

A: To use implicit differentiation, you need to follow these steps:

1. Differentiate both sides of the equation with respect to the same variable.
2. Use the chain rule and the product rule of differentiation as needed.
3. Equate the derivatives and solve for the desired variable.

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

A: Here are some common mistakes to avoid when using implicit differentiation:

• Make sure to differentiate both sides of the equation with respect to the same variable.
• Use the chain rule and the product rule of differentiation as needed.
• Equate the derivatives and solve for the desired variable.
• Check your work by plugging in the solution back into the original equation.

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

A: To check your work when using implicit differentiation, you need to plug in the solution back into the original equation. If the solution satisfies the original equation, then you have found the correct derivative.

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

A: Implicit differentiation has many real-world applications in mathematics and science. Here are a few examples:

• Finding the derivative of an implicitly defined function
• Solving optimization problems
• Finding the equation of a tangent line to a curve
• Finding the equation of a normal line to a curve

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 use implicit differentiation to find the derivative of a function that is defined in terms of multiple variables?

A: To use implicit differentiation to find the derivative of a function that is defined in terms of multiple variables, you need to follow these steps:

1. Differentiate both sides of the equation with respect to one of the variables.
2. Use the chain rule and the product rule of differentiation as needed.
3. Equate the derivatives and solve for the desired variable.

Q: Can I use implicit differentiation to find the derivative of a function that is defined in terms of a parameter?

A: Yes, you can use implicit differentiation to find the derivative of a function that is defined in terms of a parameter. However, you need to follow the same steps as before, and you need to treat the parameter as a constant.

Conclusion

Implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. In this article, we answered some common questions about implicit differentiation. We hope that this article has been helpful in understanding implicit differentiation and how to use it to find the derivative of an implicitly defined function.