1. Find The Derivatives.a) $f(x) = E^x\left(x^2 - 2x + 5\right$\]

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the derivatives of complex functions, specifically the function $f(x) = e^x\left(x^2 - 2x + 5\right)$.

The Product Rule

To find the derivative of the given function, we will use the product rule, which 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:

$$f'(x) = u'(x)v(x) + u(x)v'(x)$$

In our case, we have $u(x) = e^x$ and $v(x) = x^2 - 2x + 5$. We will first find the derivatives of $u(x)$ and $v(x)$ separately.

Derivative of $e^x$

The derivative of $e^x$ is simply $e^x$, since the exponential function is its own derivative.

Derivative of $x^2 - 2x + 5$

To find the derivative of $x^2 - 2x + 5$, we will use the power rule, which states that if we have a function of the form $f(x) = x^n$, then the derivative of $f(x)$ is given by:

$$f'(x) = nx^{n-1}$$

In our case, we have $n = 2$, so the derivative of $x^2 - 2x + 5$ is:

$$\frac{d}{dx}(x^2 - 2x + 5) = 2x - 2$$

Applying 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 $f(x) = e^x\left(x^2 - 2x + 5\right)$.

$$f'(x) = u'(x)v(x) + u(x)v'(x)$$

$$f'(x) = e^x(x^2 - 2x + 5) + e^x(2x - 2)$$

$$f'(x) = e^x(x^2 - 2x + 5 + 2x - 2)$$

$$f'(x) = e^x(x^2 + 5)$$

Simplifying the Derivative

We can simplify the derivative further by combining like terms.

$$f'(x) = e^x(x^2 + 5)$$

$$f'(x) = e^x x^2 + 5e^x$$

Conclusion

In this article, we have found the derivative of the complex function $f(x) = e^x\left(x^2 - 2x + 5\right)$ using the product rule. We have also simplified the derivative to its final form. The derivative of $f(x)$ is given by:

$$f'(x) = e^x x^2 + 5e^x$$

This result can be used to solve a wide range of problems in calculus and other fields.

Example Problems

1. Find the derivative of the function $f(x) = e^x\left(x^3 - 3x^2 + 2x - 1\right)$.
2. Find the derivative of the function $f(x) = e^x\left(x^4 - 4x^3 + 3x^2 - 2x + 1\right)$.

Solution to Example Problems

1. To find the derivative of the function $f(x) = e^x\left(x^3 - 3x^2 + 2x - 1\right)$, we will use the product rule.

$$f'(x) = u'(x)v(x) + u(x)v'(x)$$

$$f'(x) = e^x(x^3 - 3x^2 + 2x - 1) + e^x(3x^2 - 6x + 2)$$

$$f'(x) = e^x(x^3 - 3x^2 + 2x - 1 + 3x^2 - 6x + 2)$$

$$f'(x) = e^x(x^3 - 6x + 1)$$

2. To find the derivative of the function $f(x) = e^x\left(x^4 - 4x^3 + 3x^2 - 2x + 1\right)$, we will use the product rule.

$$f'(x) = u'(x)v(x) + u(x)v'(x)$$

$$f'(x) = e^x(x^4 - 4x^3 + 3x^2 - 2x + 1) + e^x(4x^3 - 12x^2 + 6x - 2)$$

$$f'(x) = e^x(x^4 - 4x^3 + 3x^2 - 2x + 1 + 4x^3 - 12x^2 + 6x - 2)$$

$$f'(x) = e^x(x^4 + 6x^3 - 9x^2 + 4x - 1)$$

Conclusion

In this article, we have found the derivatives of two complex functions using the product rule. We have also simplified the derivatives to their final forms. The derivatives of the functions are given by:

$$f'(x) = e^x(x^3 - 6x + 1)$$

$$f'(x) = e^x(x^4 + 6x^3 - 9x^2 + 4x - 1)$$

Q&A: Derivatives of Complex Functions

Q: What is the derivative of the function $f(x) = e^x\left(x^2 - 2x + 5\right)$?

A: The derivative of the function $f(x) = e^x\left(x^2 - 2x + 5\right)$ is given by:

$$f'(x) = e^x x^2 + 5e^x$$

Q: How do I find the derivative of a complex function using the product rule?

A: To find the derivative of a complex function using the product rule, you need to identify the two functions that are being multiplied together. Then, you need to find the derivatives of each of these functions separately. Finally, you can apply the product rule to find the derivative of the complex function.

Q: What is the product rule?

A: The product rule is a formula that is used to find the derivative of a complex function that is the product of two functions. The product rule is given by:

$$f'(x) = u'(x)v(x) + u(x)v'(x)$$

Q: How do I apply the product rule to find the derivative of a complex function?

A: To apply the product rule, you need to identify the two functions that are being multiplied together. Then, you need to find the derivatives of each of these functions separately. Finally, you can plug these derivatives into the product rule formula to find the derivative of the complex function.

Q: What are some common mistakes to avoid when finding the derivative of a complex function?

A: Some common mistakes to avoid when finding the derivative of a complex function include:

• Forgetting to apply the product rule when the function is a product of two functions
• Not finding the derivatives of each function separately before applying the product rule
• Not simplifying the derivative after applying the product rule

Q: How do I simplify the derivative after applying the product rule?

A: To simplify the derivative after applying the product rule, you need to combine like terms and simplify the expression as much as possible.

Q: What are some real-world applications of derivatives of complex functions?

A: Derivatives of complex functions have many real-world applications, including:

• Physics: Derivatives of complex functions are used to describe the motion of objects in physics.
• Engineering: Derivatives of complex functions are used to design and optimize systems in engineering.
• Economics: Derivatives of complex functions are used to model and analyze economic systems.

Q: How do I use derivatives of complex functions to solve problems in physics?

A: To use derivatives of complex functions to solve problems in physics, you need to apply the product rule to find the derivative of the complex function. Then, you can use the derivative to describe the motion of an object in physics.

Q: How do I use derivatives of complex functions to design and optimize systems in engineering?

A: To use derivatives of complex functions to design and optimize systems in engineering, you need to apply the product rule to find the derivative of the complex function. Then, you can use the derivative to design and optimize systems in engineering.

Q: How do I use derivatives of complex functions to model and analyze economic systems?

A: To use derivatives of complex functions to model and analyze economic systems, you need to apply the product rule to find the derivative of the complex function. Then, you can use the derivative to model and analyze economic systems.

Conclusion

In this article, we have provided a comprehensive guide to derivatives of complex functions, including a step-by-step guide to finding the derivative of a complex function using the product rule. We have also answered some common questions about derivatives of complex functions and provided some real-world applications of derivatives of complex functions.