{ \begin{array}{l} h(x) = (5-6x)^5 \\ h^{\prime}(x) = ? \end{array} \}$Choose One Answer:A. ${$-6x^5 + 5x^4(5-6x)\$}$B. ${$(-6)^5\$}$C. ${$5(5-6x)^4\$}$D. ${$-30(5-6x)^4\$}$

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. In this article, we will focus on finding the derivative of a polynomial function, specifically the function $h(x) = (5-6x)^5$. We will use the power rule and the chain rule to find the derivative of this function.

The Power Rule

The power rule is a fundamental rule in calculus that states that if $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$. This rule can be applied to any polynomial function, and it is a key tool in finding derivatives.

The Chain Rule

The chain rule is another important rule in calculus that states that if $f(x) = g(h(x))$, then $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$. This rule is used to find the derivative of composite functions.

Finding the Derivative of $h(x)$

To find the derivative of $h(x) = (5-6x)^5$, we will use the chain rule. We can rewrite $h(x)$ as $f(g(x))$, where $f(x) = x^5$ and $g(x) = 5-6x$. Then, we can find the derivative of $f(x)$ and $g(x)$ separately.

Finding the Derivative of $f(x)$

Using the power rule, we can find the derivative of $f(x) = x^5$ as follows:

$$f^{\prime}(x) = 5x^{5-1} = 5x^4$$

Finding the Derivative of $g(x)$

To find the derivative of $g(x) = 5-6x$, we can use the power rule again:

$$g^{\prime}(x) = -6$$

Applying the Chain Rule

Now that we have found the derivatives of $f(x)$ and $g(x)$, we can apply the chain rule to find the derivative of $h(x)$:

$$h^{\prime}(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x) = 5(5-6x)^4 \cdot (-6)$$

Simplifying the expression, we get:

$$h^{\prime}(x) = -30(5-6x)^4$$

Conclusion

In this article, we have used the power rule and the chain rule to find the derivative of the polynomial function $h(x) = (5-6x)^5$. We have shown that the derivative of this function is $h^{\prime}(x) = -30(5-6x)^4$. This result can be used to find the rate of change of the function with respect to its input.

Answer

The correct answer is:

• D. $-30(5-6x)^4$

Final Answer

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Introduction

In our previous article, we discussed how to find the derivative of a polynomial function using the power rule and the chain rule. In this article, we will provide a Q&A guide to help you understand the concepts and apply them to different problems.

Q: What is the derivative of a polynomial function?

A: The derivative of a polynomial function is a measure of how the function changes as its input changes. It represents the rate of change of the function with respect to its input.

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

A: To find the derivative of a polynomial function, you can use the power rule and the chain rule. The power rule states that if $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$. The chain rule states that if $f(x) = g(h(x))$, then $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$.

Q: What is the power rule?

A: The power rule is a fundamental rule in calculus that states that if $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$. This rule can be applied to any polynomial function.

Q: What is the chain rule?

A: The chain rule is another important rule in calculus that states that if $f(x) = g(h(x))$, then $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$. This rule is used to find the derivative of composite functions.

Q: How do I apply the chain rule?

A: To apply the chain rule, you need to identify the outer function and the inner function. Then, you need to find the derivative of the outer function and the inner function separately. Finally, you need to multiply the derivatives of the outer and inner functions.

Q: What is the derivative of $h(x) = (5-6x)^5$?

A: To find the derivative of $h(x) = (5-6x)^5$, we can use the chain rule. We can rewrite $h(x)$ as $f(g(x))$, where $f(x) = x^5$ and $g(x) = 5-6x$. Then, we can find the derivative of $f(x)$ and $g(x)$ separately.

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

A: Using the power rule, we can find the derivative of $f(x) = x^5$ as follows:

$$f^{\prime}(x) = 5x^{5-1} = 5x^4$$

Q: What is the derivative of $g(x) = 5-6x$?

A: To find the derivative of $g(x) = 5-6x$, we can use the power rule again:

$$g^{\prime}(x) = -6$$

Q: What is the derivative of $h(x) = (5-6x)^5$?

A: Now that we have found the derivatives of $f(x)$ and $g(x)$, we can apply the chain rule to find the derivative of $h(x)$:

$$h^{\prime}(x) = f^{\prime}(g(x)) \cdot g^{\prime}(x) = 5(5-6x)^4 \cdot (-6)$$

Simplifying the expression, we get:

$$h^{\prime}(x) = -30(5-6x)^4$$

Conclusion

In this article, we have provided a Q&A guide to help you understand the concepts of derivatives of polynomial functions. We have discussed the power rule and the chain rule, and we have applied them to find the derivative of the polynomial function $h(x) = (5-6x)^5$. We hope that this guide has been helpful in understanding the concepts and applying them to different problems.

Final Answer

The final answer is $\boxed{-30(5-6x)^4}$.