Find The Derivative Of $y=6x^2e^{-x}$.A) $6xe^{-x}(x+2$\]B) $12xe^{-x}(1-x$\]C) $6xe^x(2-x$\]D) $6xe^{-x}(2-x$\]

Introduction

In calculus, finding the derivative of a function is a crucial concept that helps us understand how the function changes as its input changes. However, when dealing with complex functions, such as the one given in this problem, $y=6x^2e^{-x}$, it can be challenging to find the derivative. In this article, we will walk you through the step-by-step process of finding the derivative of this complex function.

Understanding the Function

Before we dive into finding the derivative, let's take a closer look at the function $y=6x^2e^{-x}$. This function is a product of three terms: $6x^2$, $e^{-x}$, and a constant. To find the derivative, we need to apply the product rule of differentiation, 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 $f'(x) = u'(x)v(x) + u(x)v'(x)$.

Applying the Product Rule

To find the derivative of $y=6x^2e^{-x}$, we can apply the product rule by breaking down the function into two parts: $u(x) = 6x^2$ and $v(x) = e^{-x}$. We then find the derivatives of each part separately.

Derivative of $u(x) = 6x^2$

To find the derivative of $u(x) = 6x^2$, we can use the power rule of differentiation, 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 this case, $n = 2$, so the derivative of $u(x) = 6x^2$ is $u'(x) = 12x$.

Derivative of $v(x) = e^{-x}$

To find the derivative of $v(x) = e^{-x}$, we can use the chain rule of differentiation, which states that if we have a function of the form $f(x) = g(h(x))$, then the derivative of $f(x)$ is given by $f'(x) = g'(h(x)) \cdot h'(x)$. In this case, $g(x) = e^x$ and $h(x) = -x$, so the derivative of $v(x) = e^{-x}$ is $v'(x) = -e^{-x}$.

Finding the Derivative of the Function

Now that we have found the derivatives of $u(x) = 6x^2$ and $v(x) = e^{-x}$, we can apply the product rule to find the derivative of the function $y=6x^2e^{-x}$. 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 $f'(x) = u'(x)v(x) + u(x)v'(x)$. In this case, we have:

$$\frac{dy}{dx} = u'(x)v(x) + u(x)v'(x)$$

$$\frac{dy}{dx} = 12x \cdot e^{-x} + 6x^2 \cdot (-e^{-x})$$

$$\frac{dy}{dx} = 12xe^{-x} - 6x^2e^{-x}$$

$$\frac{dy}{dx} = 6xe^{-x}(2 - x)$$

Conclusion

In this article, we have walked you through the step-by-step process of finding the derivative of the complex function $y=6x^2e^{-x}$. We have applied the product rule of differentiation and used the power rule and chain rule to find the derivatives of the individual parts of the function. The final answer is $\boxed{6xe^{-x}(2 - x)}$.

Final Answer

The final answer is $\boxed{6xe^{-x}(2 - x)}$.

Introduction

In our previous article, we walked you through the step-by-step process of finding the derivative of the complex function $y=6x^2e^{-x}$. However, we understand that sometimes, a simple explanation is not enough, and you may have questions about the process. In this article, we will address some of the most frequently asked questions about finding the derivative of this function.

Q: What is the product rule of differentiation?

A: The product rule of differentiation is a fundamental concept in calculus that helps us find the derivative of a function that is a product of two or more functions. It 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 $f'(x) = u'(x)v(x) + u(x)v'(x)$.

Q: How do I apply the product rule to find the derivative of $y=6x^2e^{-x}$?

A: To apply the product rule, we need to break down the function into two parts: $u(x) = 6x^2$ and $v(x) = e^{-x}$. We then find the derivatives of each part separately using the power rule and chain rule. Finally, we apply the product rule to find the derivative of the function.

Q: What is the power rule of differentiation?

A: The power rule of differentiation is a fundamental concept in calculus that helps us find the derivative of a function that is a power of $x$. It states that if we have a function of the form $f(x) = x^n$, then the derivative of $f(x)$ is given by $f'(x) = nx^{n-1}$.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation is a fundamental concept in calculus that helps us find the derivative of a function that is a composition of two or more functions. It states that if we have a function of the form $f(x) = g(h(x))$, then the derivative of $f(x)$ is given by $f'(x) = g'(h(x)) \cdot h'(x)$.

Q: How do I find the derivative of $v(x) = e^{-x}$?

A: To find the derivative of $v(x) = e^{-x}$, we can use the chain rule of differentiation. We let $g(x) = e^x$ and $h(x) = -x$, so the derivative of $v(x) = e^{-x}$ is $v'(x) = -e^{-x}$.

Q: What is the final answer to the derivative of $y=6x^2e^{-x}$?

A: The final answer to the derivative of $y=6x^2e^{-x}$ is $\boxed{6xe^{-x}(2 - x)}$.

Conclusion

In this article, we have addressed some of the most frequently asked questions about finding the derivative of the complex function $y=6x^2e^{-x}$. We hope that this article has provided you with a better understanding of the product rule, power rule, and chain rule of differentiation, and how to apply them to find the derivative of this function.

Additional Resources

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Final Answer

The final answer is $\boxed{6xe^{-x}(2 - x)}$.