Find D Y D X \frac{d Y}{d X} D X D Y ​ As A Function Of X X X .Given: Y = ( − 2 X + 11 ) 5 Y=(-2 X+11)^5 Y = ( − 2 X + 11 ) 5 A) D Y D X = 5 ( − 2 X + 11 ) 4 \frac{d Y}{d X}=5(-2 X+11)^4 D X D Y ​ = 5 ( − 2 X + 11 ) 4 B) D Y D X = − 10 X 4 \frac{d Y}{d X}=-10 X^4 D X D Y ​ = − 10 X 4 C) D Y D X = − 10 ( − 2 X + 11 ) 4 \frac{d Y}{d X}=-10(-2 X+11)^4 D X D Y ​ = − 10 ( − 2 X + 11 ) 4 D) $\frac{d Y}{d X}=-2(-2

Introduction

In this article, we will explore the concept of differentiation and how it can be applied to a polynomial function. We will use the given function $y=(-2 x+11)^5$ and find its derivative as a function of $x$. This will involve using the chain rule and other differentiation techniques.

The Chain Rule

The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. It states that if we have a function of the form $y=f(g(x))$, then the derivative of $y$ with respect to $x$ is given by:

$$\frac{d y}{d x}=\frac{d f}{d g}\cdot\frac{d g}{d x}$$

Applying the Chain Rule

In our case, we have the function $y=(-2 x+11)^5$. To find its derivative, we can use the chain rule. Let's define $u=-2 x+11$ and $y=u^5$. Then, we can write:

$$y=(-2 x+11)^5=u^5$$

Now, we can use the chain rule to find the derivative of $y$ with respect to $x$:

$$\frac{d y}{d x}=\frac{d u^5}{d u}\cdot\frac{d u}{d x}$$

Finding the Derivative of $u^5$

To find the derivative of $u^5$, we can use the power rule, which states that if we have a function of the form $y=x^n$, then the derivative of $y$ with respect to $x$ is given by:

$$\frac{d y}{d x}=n x^{n-1}$$

In our case, we have $y=u^5$, so we can write:

$$\frac{d u^5}{d u}=5 u^4$$

Finding the Derivative of $u$

To find the derivative of $u$, we can use the fact that $u=-2 x+11$. Then, we can write:

$$\frac{d u}{d x}=-2$$

Putting it all Together

Now that we have found the derivatives of $u^5$ and $u$, we can use the chain rule to find the derivative of $y$ with respect to $x$:

$$\frac{d y}{d x}=\frac{d u^5}{d u}\cdot\frac{d u}{d x}=5 u^4\cdot(-2)$$

Substituting $u=-2 x+11$, we get:

$$\frac{d y}{d x}=5(-2 x+11)^4\cdot(-2)$$

Simplifying, we get:

$$\frac{d y}{d x}=-10(-2 x+11)^4$$

Conclusion

In this article, we have used the chain rule and other differentiation techniques to find the derivative of the function $y=(-2 x+11)^5$. We have shown that the derivative of $y$ with respect to $x$ is given by:

$$\frac{d y}{d x}=-10(-2 x+11)^4$$

This result is consistent with option C.

Final Answer

The final answer is:

• $\boxed{-10(-2 x+11)^4}$
  Q&A: Differentiation of a Polynomial Function =====================================================

Introduction

In our previous article, we explored the concept of differentiation and how it can be applied to a polynomial function. We used the chain rule and other differentiation techniques to find the derivative of the function $y=(-2 x+11)^5$. In this article, we will answer some common questions related to differentiation of a polynomial function.

Q: What is the chain rule?

A: The chain rule is a fundamental concept in calculus that allows us to differentiate composite functions. It states that if we have a function of the form $y=f(g(x))$, then the derivative of $y$ with respect to $x$ is given by:

$$\frac{d y}{d x}=\frac{d f}{d g}\cdot\frac{d g}{d x}$$

Q: How do I apply the chain rule?

A: To apply the chain rule, you need to identify the inner and outer functions. The inner function is the function that is being raised to a power, and the outer function is the function that is being applied to the inner function. Then, you can use the chain rule to find the derivative of the outer function with respect to the inner function.

Q: What is the power rule?

A: The power rule is a rule in calculus that states that if we have a function of the form $y=x^n$, then the derivative of $y$ with respect to $x$ is given by:

$$\frac{d y}{d x}=n x^{n-1}$$

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. First, identify the terms in the polynomial function and their corresponding powers. Then, use the power rule to find the derivative of each term. Finally, use the chain rule to find the derivative of the composite function.

Q: What is the derivative of $y=x^2+3x-4$?

A: To find the derivative of $y=x^2+3x-4$, we can use the power rule and the sum rule. The derivative of $x^2$ is $2x$, the derivative of $3x$ is $3$, and the derivative of $-4$ is $0$. Therefore, the derivative of $y=x^2+3x-4$ is:

$$\frac{d y}{d x}=2x+3$$

Q: What is the derivative of $y=(2x-3)^4$?

A: To find the derivative of $y=(2x-3)^4$, we can use the chain rule. Let's define $u=2x-3$ and $y=u^4$. Then, we can write:

$$\frac{d y}{d x}=\frac{d u^4}{d u}\cdot\frac{d u}{d x}$$

The derivative of $u^4$ is $4u^3$, and the derivative of $u$ is $2$. Therefore, the derivative of $y=(2x-3)^4$ is:

$$\frac{d y}{d x}=4u^3\cdot2=8u^3$$

Substituting $u=2x-3$, we get:

$$\frac{d y}{d x}=8(2x-3)^3$$

Conclusion

In this article, we have answered some common questions related to differentiation of a polynomial function. We have used the chain rule and other differentiation techniques to find the derivatives of various polynomial functions. We hope that this article has been helpful in understanding the concept of differentiation and how it can be applied to polynomial functions.

Final Answer

The final answer is:

• $\boxed{8(2x-3)^3}$