Find The Derivative With Respect To X X X If F ( X ) = ( 12 X 2 − 24 X + 24 ) E X F(x)=\left(12 X^2-24 X+24\right) E^x F ( X ) = ( 12 X 2 − 24 X + 24 ) E X .The Derivative With Respect To X X X Is F ′ ( X ) = F^{\prime}(x) = F ′ ( X ) = □ \square □ .

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Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. In this article, we will focus on finding the derivative with respect to $x$ of a given function, $f(x)=\left(12 x^2-24 x+24\right) e^x$. This function is a product of two functions, and we will use the product rule of differentiation to find its derivative.

The Product Rule of Differentiation

The product rule of differentiation states that if we have a function of the form $f(x) = u(x)v(x)$, where $u(x)$ and $v(x)$ are both functions of $x$, then the derivative of $f(x)$ with respect to $x$ is given by:

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

Finding the Derivative of the Given Function

To find the derivative of the given function, $f(x)=\left(12 x^2-24 x+24\right) e^x$, we can use the product rule of differentiation. Let $u(x) = 12 x^2-24 x+24$ and $v(x) = e^x$. Then, we can find the derivatives of $u(x)$ and $v(x)$ with respect to $x$ as follows:

$$u^{\prime}(x) = \frac{d}{dx} (12 x^2-24 x+24) = 24 x - 24$$

$$v^{\prime}(x) = \frac{d}{dx} (e^x) = e^x$$

Applying the Product Rule

Now that we have found the derivatives of $u(x)$ and $v(x)$ with respect to $x$, we can apply the product rule of differentiation to find the derivative of $f(x)$ with respect to $x$:

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

$$f^{\prime}(x) = (24 x - 24) e^x + (12 x^2-24 x+24) e^x$$

Simplifying the Derivative

We can simplify the derivative of $f(x)$ with respect to $x$ by combining like terms:

$$f^{\prime}(x) = (24 x - 24 + 12 x^2-24 x+24) e^x$$

$$f^{\prime}(x) = (12 x^2) e^x$$

Conclusion

In this article, we have found the derivative with respect to $x$ of a given function, $f(x)=\left(12 x^2-24 x+24\right) e^x$. We used the product rule of differentiation to find the derivative of the function, and then simplified the result to obtain the final answer.

Final Answer

The derivative with respect to $x$ of the given function is:

$$f^{\prime}(x) = (12 x^2) e^x$$

Example Use Case

The derivative of a function can be used to find the rate of change of the function with respect to one of its variables. For example, if we have a function that represents the position of an object with respect to time, the derivative of the function can be used to find the velocity of the object.

Step-by-Step Solution

To find the derivative of the given function, we can follow these steps:

1. Identify the two functions $u(x)$ and $v(x)$ that make up the given function.
2. Find the derivatives of $u(x)$ and $v(x)$ with respect to $x$.
3. Apply the product rule of differentiation to find the derivative of the given function.
4. Simplify the result to obtain the final answer.

Common Mistakes

When finding the derivative of a function, it is common to make mistakes such as:

• Forgetting to apply the product rule of differentiation.
• Not simplifying the result correctly.
• Making errors when finding the derivatives of the individual functions.

Tips and Tricks

To find the derivative of a function, it is helpful to:

• Use the product rule of differentiation when the function is a product of two functions.
• Simplify the result carefully to avoid making mistakes.
• Check your work by plugging in values of $x$ and checking that the result is correct.

Related Topics

The derivative of a function is related to other topics in calculus, such as:

• The chain rule of differentiation.
• The quotient rule of differentiation.
• The fundamental theorem of calculus.

Conclusion

In this article, we have found the derivative with respect to $x$ of a given function, $f(x)=\left(12 x^2-24 x+24\right) e^x$. We used the product rule of differentiation to find the derivative of the function, and then simplified the result to obtain the final answer.

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Introduction

In our previous article, we found the derivative with respect to $x$ of a given function, $f(x)=\left(12 x^2-24 x+24\right) e^x$. In this article, we will answer some common questions related to finding the derivative of a function.

Q1: What is the product rule of differentiation?

A1: The product rule of differentiation is a rule that is used to find the derivative of a function that is a product of two functions. It states that if we have a function of the form $f(x) = u(x)v(x)$, where $u(x)$ and $v(x)$ are both functions of $x$, then the derivative of $f(x)$ with respect to $x$ is given by:

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

Q2: How do I find the derivative of a function that is a product of two functions?

A2: To find the derivative of a function that is a product of two functions, you can use the product rule of differentiation. First, identify the two functions $u(x)$ and $v(x)$ that make up the given function. Then, find the derivatives of $u(x)$ and $v(x)$ with respect to $x$. Finally, apply the product rule of differentiation to find the derivative of the given function.

Q3: What is the chain rule of differentiation?

A3: The chain rule of differentiation is a rule that is used to find the derivative of a composite function. A composite function is a function that is composed of two or more functions. The chain rule states that if we have a composite function of the form $f(x) = g(h(x))$, where $g(x)$ and $h(x)$ are both functions of $x$, then the derivative of $f(x)$ with respect to $x$ is given by:

$$f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$$

Q4: How do I find the derivative of a composite function?

A4: To find the derivative of a composite function, you can use the chain rule of differentiation. First, identify the two functions $g(x)$ and $h(x)$ that make up the composite function. Then, find the derivatives of $g(x)$ and $h(x)$ with respect to $x$. Finally, apply the chain rule of differentiation to find the derivative of the composite function.

Q5: What is the quotient rule of differentiation?

A5: The quotient rule of differentiation is a rule that is used to find the derivative of a function that is a quotient of two functions. It states that if we have a function of the form $f(x) = \frac{u(x)}{v(x)}$, where $u(x)$ and $v(x)$ are both functions of $x$, then the derivative of $f(x)$ with respect to $x$ is given by:

$$f^{\prime}(x) = \frac{u^{\prime}(x)v(x) - u(x)v^{\prime}(x)}{v(x)^2}$$

Q6: How do I find the derivative of a function that is a quotient of two functions?

A6: To find the derivative of a function that is a quotient of two functions, you can use the quotient rule of differentiation. First, identify the two functions $u(x)$ and $v(x)$ that make up the given function. Then, find the derivatives of $u(x)$ and $v(x)$ with respect to $x$. Finally, apply the quotient rule of differentiation to find the derivative of the given function.

Q7: What is the fundamental theorem of calculus?

A7: The fundamental theorem of calculus is a theorem that relates the derivative of a function to the definite integral of the function. It states that if we have a function $f(x)$ and a constant $c$, then the definite integral of $f(x)$ from $a$ to $b$ is given by:

$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$

where $F(x)$ is the antiderivative of $f(x)$.

Q8: How do I use the fundamental theorem of calculus?

A8: To use the fundamental theorem of calculus, you can first find the antiderivative of the given function. Then, evaluate the antiderivative at the upper and lower limits of integration. Finally, subtract the value of the antiderivative at the lower limit of integration from the value of the antiderivative at the upper limit of integration.

Conclusion

In this article, we have answered some common questions related to finding the derivative of a function. We have discussed the product rule of differentiation, the chain rule of differentiation, the quotient rule of differentiation, and the fundamental theorem of calculus. We hope that this article has been helpful in answering your questions and providing you with a better understanding of the derivative of a function.