Differentiate The Following Function.${ F(x) = X^5 E^{7x} }$ { F^{\prime}(x) = \}

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Introduction

In this article, we will differentiate the given function $f(x) = x^5 e^{7x}$ using the product rule of differentiation. The product rule states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative of $f(x)$ is given by $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.

Step 1: Identify the Functions $u(x)$ and $v(x)$

In the given function $f(x) = x^5 e^{7x}$, we can identify $u(x) = x^5$ and $v(x) = e^{7x}$.

Step 2: Find the Derivatives of $u(x)$ and $v(x)$

To find the derivative of $u(x) = x^5$, we use the power rule of differentiation, which states that if $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$. Therefore, the derivative of $u(x) = x^5$ is $u^{\prime}(x) = 5x^4$.

To find the derivative of $v(x) = e^{7x}$, we use the chain rule of differentiation, which states that if $f(x) = e^{u(x)}$, then $f^{\prime}(x) = e^{u(x)}u^{\prime}(x)$. In this case, $u(x) = 7x$, so the derivative of $v(x) = e^{7x}$ is $v^{\prime}(x) = 7e^{7x}$.

Step 3: Apply the Product Rule

Now that we have found the derivatives of $u(x)$ and $v(x)$, we can apply the product rule to find the derivative of $f(x) = x^5 e^{7x}$. The product rule states that if $f(x) = u(x)v(x)$, then $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$. Therefore, the derivative of $f(x) = x^5 e^{7x}$ is:

$$f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$$

$$f^{\prime}(x) = 5x^4 e^{7x} + x^5 7e^{7x}$$

Simplifying the Derivative

We can simplify the derivative by combining like terms:

$$f^{\prime}(x) = 5x^4 e^{7x} + 7x^5 e^{7x}$$

$$f^{\prime}(x) = (5x^4 + 7x^5) e^{7x}$$

Conclusion

In this article, we have differentiated the function $f(x) = x^5 e^{7x}$ using the product rule of differentiation. We identified the functions $u(x)$ and $v(x)$, found their derivatives, and applied the product rule to find the derivative of $f(x)$. The final derivative is $f^{\prime}(x) = (5x^4 + 7x^5) e^{7x}$.

Example Use Case

The derivative of $f(x) = x^5 e^{7x}$ can be used to find the maximum or minimum of the function. For example, if we want to find the maximum of $f(x)$, we can set $f^{\prime}(x) = 0$ and solve for $x$. This will give us the critical points of the function, which can be used to determine the maximum or minimum.

Code Implementation

The derivative of $f(x) = x^5 e^{7x}$ can be implemented in code using a programming language such as Python or MATLAB. For example, in Python, we can use the sympy library to define the function and its derivative:

import sympy as sp
x = sp.symbols('x')
f = x**5 * sp.exp(7*x)
f_prime = sp.diff(f, x)
print(f_prime)

This code will output the derivative of $f(x) = x^5 e^{7x}$, which is $(5x^4 + 7x^5) e^{7x}$.

Conclusion

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Q: What is the product rule of differentiation?

A: The product rule of differentiation states that if we have a function of the form $f(x) = u(x)v(x)$, then the derivative of $f(x)$ is given by $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$.

Q: How do I apply the product rule to find the derivative of $f(x) = x^5 e^{7x}$?

A: To apply the product rule, we need to identify the functions $u(x)$ and $v(x)$, find their derivatives, and then use the product rule formula to find the derivative of $f(x)$. In this case, we have $u(x) = x^5$ and $v(x) = e^{7x}$. We can find their derivatives using the power rule and chain rule, respectively.

Q: What is the derivative of $u(x) = x^5$?

A: The derivative of $u(x) = x^5$ is $u^{\prime}(x) = 5x^4$.

Q: What is the derivative of $v(x) = e^{7x}$?

A: The derivative of $v(x) = e^{7x}$ is $v^{\prime}(x) = 7e^{7x}$.

Q: How do I use the product rule to find the derivative of $f(x) = x^5 e^{7x}$?

A: To use the product rule, we need to plug in the derivatives of $u(x)$ and $v(x)$ into the product rule formula:

$$f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$$

$$f^{\prime}(x) = 5x^4 e^{7x} + x^5 7e^{7x}$$

Q: Can I simplify the derivative of $f(x) = x^5 e^{7x}$?

A: Yes, we can simplify the derivative by combining like terms:

$$f^{\prime}(x) = 5x^4 e^{7x} + 7x^5 e^{7x}$$

$$f^{\prime}(x) = (5x^4 + 7x^5) e^{7x}$$

Q: What is the final derivative of $f(x) = x^5 e^{7x}$?

A: The final derivative of $f(x) = x^5 e^{7x}$ is $f^{\prime}(x) = (5x^4 + 7x^5) e^{7x}$.

Q: How can I use the derivative of $f(x) = x^5 e^{7x}$ in real-world applications?

A: The derivative of $f(x) = x^5 e^{7x}$ can be used to find the maximum or minimum of the function. For example, if we want to find the maximum of $f(x)$, we can set $f^{\prime}(x) = 0$ and solve for $x$. This will give us the critical points of the function, which can be used to determine the maximum or minimum.

Q: Can I implement the derivative of $f(x) = x^5 e^{7x}$ in code?

A: Yes, we can implement the derivative of $f(x) = x^5 e^{7x}$ in code using a programming language such as Python or MATLAB. For example, in Python, we can use the sympy library to define the function and its derivative:

import sympy as sp
x = sp.symbols('x')
f = x**5 * sp.exp(7*x)
f_prime = sp.diff(f, x)
print(f_prime)

This code will output the derivative of $f(x) = x^5 e^{7x}$, which is $(5x^4 + 7x^5) e^{7x}$.

Conclusion

In this Q&A article, we have discussed the product rule of differentiation and how to apply it to find the derivative of $f(x) = x^5 e^{7x}$. We have also provided examples and code implementations to illustrate the concept.