Find The Derivative Of $g(x) = \left(3x^2 - 5x + 2\right) E^x$.$g'(x) =$

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Introduction

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In calculus, the derivative of a function represents the rate of change of the function with respect to its input. When dealing with complex functions, finding the derivative can be a challenging task. In this article, we will explore how to find the derivative of a function that involves both polynomial and exponential terms.

The Function to Differentiate

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The function we will be differentiating is given by:

$$g(x) = \left(3x^2 - 5x + 2\right) e^x$$

This function involves both polynomial and exponential terms. To find the derivative, we will need to apply the product rule of differentiation.

The Product Rule of Differentiation

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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 the function is given by:

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

In our case, we have:

$$g(x) = \left(3x^2 - 5x + 2\right) e^x$$

We can identify the two functions $u(x)$ and $v(x)$ as:

$$u(x) = 3x^2 - 5x + 2$$

$$v(x) = e^x$$

Finding the Derivatives of $u(x)$ and $v(x)$

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To apply the product rule, we need to find the derivatives of $u(x)$ and $v(x)$.

The derivative of $u(x)$ is given by:

$$u'(x) = \frac{d}{dx} \left(3x^2 - 5x + 2\right)$$

$$u'(x) = 6x - 5$$

The derivative of $v(x)$ is given by:

$$v'(x) = \frac{d}{dx} \left(e^x\right)$$

$$v'(x) = e^x$$

Applying the Product Rule

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Now that we have found the derivatives of $u(x)$ and $v(x)$, we can apply the product rule to find the derivative of $g(x)$.

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

$$g'(x) = (6x - 5)e^x + (3x^2 - 5x + 2)e^x$$

Simplifying the Derivative

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We can simplify the derivative by combining like terms.

$$g'(x) = (6x - 5 + 3x^2 - 5x + 2)e^x$$

$$g'(x) = (3x^2 - 9x - 3)e^x$$

Conclusion

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In this article, we have shown how to find the derivative of a complex function that involves both polynomial and exponential terms. We applied the product rule of differentiation to find the derivative of the function, and then simplified the result to obtain the final answer.

Final Answer

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The derivative of the function $g(x) = \left(3x^2 - 5x + 2\right) e^x$ is given by:

$$g'(x) = (3x^2 - 9x - 3)e^x$$

This is the final answer to the problem.

Example Use Cases

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The derivative of a function can be used in a variety of applications, including:

• Physics: The derivative of a function can be used to model the motion of an object, including its velocity and acceleration.
• Engineering: The derivative of a function can be used to design and optimize systems, including electrical and mechanical systems.
• Economics: The derivative of a function can be used to model the behavior of economic systems, including the behavior of supply and demand.

Tips and Tricks

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When differentiating complex functions, it is often helpful to:

• Break down the function into simpler components: This can make it easier to apply the product rule and other differentiation rules.
• Use the chain rule: This can be useful when differentiating composite functions.
• Check your work: This can help you catch any mistakes you may have made in the differentiation process.

Common Mistakes

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When differentiating complex functions, it is easy to make mistakes. Some common mistakes include:

• Forgetting to apply the product rule: This can result in an incorrect derivative.
• Making mistakes when differentiating individual components: This can result in an incorrect derivative.
• Not checking your work: This can result in an incorrect derivative.

Conclusion

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In conclusion, finding the derivative of a complex function can be a challenging task. However, by applying the product rule and other differentiation rules, we can find the derivative of even the most complex functions. Remember to break down the function into simpler components, use the chain rule, and check your work to ensure that you get the correct answer.

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Introduction

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In our previous article, we explored how to find the derivative of a complex function that involves both polynomial and exponential terms. In this article, we will answer some common questions that readers may have about finding the derivative of a complex function.

Q: What is the product rule of differentiation?

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A: The product rule of differentiation is a rule that allows us to find the derivative of a function that is the product of two or more functions. The product rule states that if we have a function of the form:

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

then the derivative of the function is given by:

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

Q: How do I apply the product rule?

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A: To apply the product rule, you need to identify the two functions $u(x)$ and $v(x)$ that make up the product. Then, you need to find the derivatives of $u(x)$ and $v(x)$, and plug these derivatives into the product rule formula.

Q: What is the chain rule of differentiation?

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A: The chain rule of differentiation is a rule that allows us to find the derivative of a composite function. A composite function is a function that is the result of applying one or more functions to another function. The chain rule states that if we have a composite function of the form:

$$f(x) = g(h(x))$$

then the derivative of the function is given by:

$$f'(x) = g'(h(x)) \cdot h'(x)$$

Q: How do I apply the chain rule?

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A: To apply the chain rule, you need to identify the outer function $g(x)$ and the inner function $h(x)$. Then, you need to find the derivatives of $g(x)$ and $h(x)$, and plug these derivatives into the chain rule formula.

Q: What is the derivative of $e^x$?

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A: The derivative of $e^x$ is $e^x$ itself. This is because the exponential function is its own derivative.

Q: What is the derivative of $\ln x$?

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A: The derivative of $\ln x$ is $\frac{1}{x}$. This is because the natural logarithm function is the inverse of the exponential function.

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

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A: To find the derivative of a trigonometric function, you need to use the following formulas:

• $\frac{d}{dx} \sin x = \cos x$
• $\frac{d}{dx} \cos x = -\sin x$
• $\frac{d}{dx} \tan x = \sec^2 x$
• $\frac{d}{dx} \csc x = -\csc x \cot x$
• $\frac{d}{dx} \sec x = \sec x \tan x$
• $\frac{d}{dx} \cot x = -\csc^2 x$

Q: What is the derivative of a rational function?

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A: The derivative of a rational function is given by the following formula:

$$\frac{d}{dx} \frac{f(x)}{g(x)} = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$$

Q: How do I find the derivative of a function with multiple variables?

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A: To find the derivative of a function with multiple variables, you need to use the following formulas:

• $\frac{\partial}{\partial x} f(x,y) = \frac{\partial f}{\partial x}$
• $\frac{\partial}{\partial y} f(x,y) = \frac{\partial f}{\partial y}$

Conclusion

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In conclusion, finding the derivative of a complex function can be a challenging task. However, by applying the product rule, chain rule, and other differentiation rules, we can find the derivative of even the most complex functions. Remember to break down the function into simpler components, use the chain rule, and check your work to ensure that you get the correct answer.

Tips and Tricks

---

When differentiating complex functions, it is often helpful to:

• Break down the function into simpler components: This can make it easier to apply the product rule and other differentiation rules.
• Use the chain rule: This can be useful when differentiating composite functions.
• Check your work: This can help you catch any mistakes you may have made in the differentiation process.

Common Mistakes

---

When differentiating complex functions, it is easy to make mistakes. Some common mistakes include:

• Forgetting to apply the product rule: This can result in an incorrect derivative.
• Making mistakes when differentiating individual components: This can result in an incorrect derivative.
• Not checking your work: This can result in an incorrect derivative.

Conclusion

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In conclusion, finding the derivative of a complex function requires a good understanding of the product rule, chain rule, and other differentiation rules. By applying these rules and checking your work, you can find the derivative of even the most complex functions. Remember to break down the function into simpler components, use the chain rule, and check your work to ensure that you get the correct answer.