Given Y = 5 E X + 6 X 20 13 Y=\sqrt[13]{5 E^x+6 X^{20}} Y = 13 5 E X + 6 X 20 ​ , Find D Y D X \frac{d Y}{d X} D X D Y ​ .

$$

Introduction

Implicit differentiation is a powerful technique used 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 the given function $y=\sqrt[13]{5 e^x+6 x^{20}}$. We will break down the process into manageable steps and provide a clear explanation of each step.

Step 1: Understand the Problem

The given function is $y=\sqrt[13]{5 e^x+6 x^{20}}$. We are asked to find the derivative of this function with respect to $x$, denoted as $\frac{d y}{d x}$. To do this, we will use the technique of implicit differentiation.

Step 2: Apply the Chain Rule

The chain rule is a fundamental rule in calculus that states that if we have a composite function of the form $f(g(x))$, then the derivative of this function is given by $f'(g(x)) \cdot g'(x)$. In our case, we have a composite function of the form $y=\sqrt[13]{5 e^x+6 x^{20}}$. We can rewrite this function as $y=(5 e^x+6 x^{20})^{\frac{1}{13}}$. Now, we can apply the chain rule to find the derivative of this function.

Step 3: Differentiate the Outer Function

The outer function is $y=(5 e^x+6 x^{20})^{\frac{1}{13}}$. We can differentiate this function using the power rule, which states that if we have a function of the form $f(x)^n$, then the derivative of this function is given by $n \cdot f(x)^{n-1} \cdot f'(x)$. In our case, we have $n=\frac{1}{13}$ and $f(x)=5 e^x+6 x^{20}$. Therefore, the derivative of the outer function is given by:

$$\frac{d y}{d x} = \frac{1}{13} \cdot (5 e^x+6 x^{20})^{\frac{1}{13}-1} \cdot \frac{d}{d x} (5 e^x+6 x^{20})$$

Step 4: Differentiate the Inner Function

The inner function is $5 e^x+6 x^{20}$. We can differentiate this function using the sum rule, which states that if we have a function of the form $f(x)+g(x)$, then the derivative of this function is given by $f'(x)+g'(x)$. In our case, we have $f(x)=5 e^x$ and $g(x)=6 x^{20}$. Therefore, the derivative of the inner function is given by:

$$\frac{d}{d x} (5 e^x+6 x^{20}) = 5 e^x+6 \cdot 20 x^{19}$$

Step 5: Substitute the Derivative of the Inner Function

Now that we have found the derivative of the inner function, we can substitute it into the expression for the derivative of the outer function. We get:

$$\frac{d y}{d x} = \frac{1}{13} \cdot (5 e^x+6 x^{20})^{\frac{1}{13}-1} \cdot (5 e^x+120 x^{19})$$

Step 6: Simplify the Expression

We can simplify the expression for the derivative of the outer function by combining like terms. We get:

$$\frac{d y}{d x} = \frac{1}{13} \cdot (5 e^x+6 x^{20})^{-\frac{12}{13}} \cdot (5 e^x+120 x^{19})$$

Conclusion

In this article, we used the technique of implicit differentiation to find the derivative of the given function $y=\sqrt[13]{5 e^x+6 x^{20}}$. We broke down the process into manageable steps and provided a clear explanation of each step. We applied the chain rule, differentiated the outer function, differentiated the inner function, substituted the derivative of the inner function, and simplified the expression. The final answer is:

$$\frac{d y}{d x} = \frac{1}{13} \cdot (5 e^x+6 x^{20})^{-\frac{12}{13}} \cdot (5 e^x+120 x^{19})$$

Example Use Case

Implicit differentiation is a powerful technique that can be used to find the derivative of an implicitly defined function. One example use case is in physics, where we may need to find the derivative of a function that describes the motion of an object. For instance, if we have a function that describes the position of an object as a function of time, we can use implicit differentiation to find the velocity of the object.

Code Implementation

Here is an example code implementation in Python:

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

y = (5 * sp.exp(x) + 6 * x20)(1/13)

dy_dx = sp.diff(y, x)

print(dy_dx)

This code uses the SymPy library to define the variable, function, and derivative. The sp.diff function is used to find the derivative of the function, and the result is printed to the console.

Conclusion

In conclusion, implicit differentiation is a powerful technique that can be used to find the derivative of an implicitly defined function. We used this technique to find the derivative of the given function $y=\sqrt[13]{5 e^x+6 x^{20}}$. We broke down the process into manageable steps and provided a clear explanation of each step. We applied the chain rule, differentiated the outer function, differentiated the inner function, substituted the derivative of the inner function, and simplified the expression. The final answer is:

\frac{d y}{d x} = \frac{1}{13} \cdot (5 e^x+6 x^{20})^{-\frac{12}{13}} \cdot (5 e^x+120 x^{19})$<br/> **Implicit Differentiation: A Q&A Guide** =====================================

Introduction

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

Q: What is implicit differentiation?

A: Implicit differentiation is a technique used to find the derivative of an implicitly defined function. It involves differentiating both sides of an equation with respect to the variable, while treating the other variables as constants.

Q: When is implicit differentiation used?

A: Implicit differentiation is used when the function is defined implicitly, meaning that it is not possible to isolate the variable on one side of the equation. This is often the case in physics, engineering, and other fields where the function is defined by a system of equations.

Q: How does implicit differentiation work?

A: Implicit differentiation involves differentiating both sides of an equation with respect to the variable, while treating the other variables as constants. This is done using the chain rule and the product rule.

Q: What are some common applications of implicit differentiation?

A: Implicit differentiation has many applications in physics, engineering, and other fields. Some common applications include:

• Finding the derivative of a function that is defined implicitly
• Finding the velocity and acceleration of an object
• Finding the derivative of a function that is defined by a system of equations
• Finding the derivative of a function that involves trigonometric functions

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

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

• Failing to treat the other variables as constants
• Failing to use the chain rule and the product rule correctly
• Failing to simplify the expression for the derivative
• Failing to check the domain of the function

Q: How do I know if I need to use implicit differentiation?

A: You need to use implicit differentiation if the function is defined implicitly, meaning that it is not possible to isolate the variable on one side of the equation.

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

A: No, implicit differentiation is only used to find the derivative of a function that is defined implicitly.

Q: How do I find the derivative of a function that involves trigonometric functions?

A: To find the derivative of a function that involves trigonometric functions, you can use the chain rule and the product rule. You can also use the trigonometric identities to simplify the expression for the derivative.

Q: Can I use implicit differentiation to find the derivative of a function that involves logarithmic functions?

A: Yes, you can use implicit differentiation to find the derivative of a function that involves logarithmic functions. You can use the chain rule and the product rule to find the derivative.

Q: How do I check the domain of a function that is defined implicitly?

A: To check the domain of a function that is defined implicitly, you need to check the values of the variable that make the function undefined. You can do this by setting the denominator of the function equal to zero and solving for the variable.

Q: Can I use implicit differentiation to find the derivative of a function that involves rational functions?

A: Yes, you can use implicit differentiation to find the derivative of a function that involves rational functions. You can use the chain rule and the product rule to find the derivative.

Conclusion

Implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. It involves differentiating both sides of an equation with respect to the variable, while treating the other variables as constants. By following the steps outlined in this article, you can use implicit differentiation to find the derivative of a function that is defined implicitly.

Example Use Case

Implicit differentiation has many applications in physics, engineering, and other fields. For example, if we have a function that describes the position of an object as a function of time, we can use implicit differentiation to find the velocity and acceleration of the object.

Code Implementation

Here is an example code implementation in Python:

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the function
y = (5 * sp.exp(x) + 6 * x**20)**(1/13)

# Find the derivative of the function
dy_dx = sp.diff(y, x)

# Print the derivative
print(dy_dx)

This code uses the SymPy library to define the variable, function, and derivative. The sp.diff function is used to find the derivative of the function, and the result is printed to the console.

Conclusion

In conclusion, implicit differentiation is a powerful technique used to find the derivative of an implicitly defined function. By following the steps outlined in this article, you can use implicit differentiation to find the derivative of a function that is defined implicitly.