Find The Derivative Of F ( X ) = 2 X 4 − 5 X 3 + 3 F(x) = 2x^4 - 5x^3 + 3 F ( X ) = 2 X 4 − 5 X 3 + 3 .A. 8 X 3 − 15 X 2 8x^3 - 15x^2 8 X 3 − 15 X 2 B. 4 X 3 − 3 X 2 + 3 4x^3 - 3x^2 + 3 4 X 3 − 3 X 2 + 3 C. 4 X 3 − 3 X 2 4x^3 - 3x^2 4 X 3 − 3 X 2 D. 8 X 3 − 15 X 2 + 3 8x^3 - 15x^2 + 3 8 X 3 − 15 X 2 + 3

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to one of its variables. In this article, we will focus on finding the derivative of a polynomial function, specifically the function $f(x) = 2x^4 - 5x^3 + 3$. The derivative of this function will be used to determine the rate of change of the function at any given point.

What is a Derivative?

A derivative is a measure of how a function changes as its input changes. It is a fundamental concept in calculus and is used to model real-world phenomena such as the motion of objects, the growth of populations, and the behavior of economic systems. The derivative of a function is denoted by the symbol $f'(x)$ and is defined as the limit of the difference quotient.

The Power Rule

To find the derivative of a polynomial function, we can use the power rule, which states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. This rule can be applied to any term in a polynomial function, and it is a powerful tool for finding derivatives.

Applying the Power Rule

To find the derivative of the function $f(x) = 2x^4 - 5x^3 + 3$, we can apply the power rule to each term in the function. The first term, $2x^4$, has a power of 4, so we multiply the coefficient, 2, by the power, 4, and subtract 1 from the power to get $8x^3$. The second term, $-5x^3$, has a power of 3, so we multiply the coefficient, -5, by the power, 3, and subtract 1 from the power to get $-15x^2$. The third term, 3, is a constant, so its derivative is 0.

Finding the Derivative

Using the power rule, we can find the derivative of the function $f(x) = 2x^4 - 5x^3 + 3$ as follows:

$f'(x) = \frac{d}{dx}(2x^4 - 5x^3 + 3)$ $f'(x) = \frac{d}{dx}(2x^4) - \frac{d}{dx}(5x^3) + \frac{d}{dx}(3)$ $f'(x) = 8x^3 - 15x^2 + 0$ $f'(x) = 8x^3 - 15x^2$

Conclusion

In this article, we found the derivative of the polynomial function $f(x) = 2x^4 - 5x^3 + 3$ using the power rule. The derivative of the function is $f'(x) = 8x^3 - 15x^2$. This result can be used to determine the rate of change of the function at any given point.

Answer

The correct answer is A. $8x^3 - 15x^2$.

Discussion

The derivative of a function is a measure of how the function changes as its input changes. It is a fundamental concept in calculus and is used to model real-world phenomena such as the motion of objects, the growth of populations, and the behavior of economic systems. The power rule is a powerful tool for finding derivatives, and it can be applied to any term in a polynomial function.

Example

Find the derivative of the function $f(x) = 3x^2 - 2x + 1$.

Solution

Using the power rule, we can find the derivative of the function $f(x) = 3x^2 - 2x + 1$ as follows:

$f'(x) = \frac{d}{dx}(3x^2 - 2x + 1)$ $f'(x) = \frac{d}{dx}(3x^2) - \frac{d}{dx}(2x) + \frac{d}{dx}(1)$ $f'(x) = 6x - 2 + 0$ $f'(x) = 6x - 2$

Practice Problems

1. Find the derivative of the function $f(x) = 2x^3 - 3x^2 + 1$.
2. Find the derivative of the function $f(x) = x^4 - 2x^3 + 3x^2$.
3. Find the derivative of the function $f(x) = 4x^2 - 5x + 2$.

Solutions

1. $f'(x) = 6x^2 - 6x$
2. $f'(x) = 4x^3 - 6x^2 + 6x$
3. $f'(x) = 8x - 5$

Conclusion

In this article, we found the derivative of a polynomial function using the power rule. The power rule is a powerful tool for finding derivatives, and it can be applied to any term in a polynomial function. We also provided examples and practice problems to help reinforce the concept of finding derivatives.

Introduction

In our previous article, we discussed how to find the derivative of a polynomial function using the power rule. In this article, we will answer some frequently asked questions about finding the derivative of a polynomial function.

Q: What is the power rule?

A: The power rule is a formula for finding the derivative of a polynomial function. It states that if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.

Q: How do I apply the power rule to a polynomial function?

A: To apply the power rule to a polynomial function, you need to identify the terms in the function and apply the power rule to each term. For example, if you have the function $f(x) = 2x^4 - 5x^3 + 3$, you would apply the power rule to each term as follows:

$f'(x) = \frac{d}{dx}(2x^4) - \frac{d}{dx}(5x^3) + \frac{d}{dx}(3)$ $f'(x) = 8x^3 - 15x^2 + 0$ $f'(x) = 8x^3 - 15x^2$

Q: What is the derivative of a constant function?

A: The derivative of a constant function is 0. This is because the derivative of a function represents the rate of change of the function, and a constant function does not change.

Q: Can I use the power rule to find the derivative of a function with a negative exponent?

A: Yes, you can use the power rule to find the derivative of a function with a negative exponent. For example, if you have the function $f(x) = 2x^{-3}$, you would apply the power rule as follows:

$f'(x) = \frac{d}{dx}(2x^{-3})$ $f'(x) = -6x^{-4}$ $f'(x) = -\frac{6}{x^4}$

Q: Can I use the power rule to find the derivative of a function with a fractional exponent?

A: Yes, you can use the power rule to find the derivative of a function with a fractional exponent. For example, if you have the function $f(x) = 2x^{\frac{1}{2}}$, you would apply the power rule as follows:

$f'(x) = \frac{d}{dx}(2x^{\frac{1}{2}})$ $f'(x) = x^{-\frac{1}{2}}$ $f'(x) = \frac{1}{\sqrt{x}}$

Q: What is the derivative of a polynomial function with multiple terms?

A: The derivative of a polynomial function with multiple terms is found by applying the power rule to each term. For example, if you have the function $f(x) = 2x^4 - 5x^3 + 3x^2$, you would apply the power rule to each term as follows:

$f'(x) = \frac{d}{dx}(2x^4) - \frac{d}{dx}(5x^3) + \frac{d}{dx}(3x^2)$ $f'(x) = 8x^3 - 15x^2 + 6x$ $f'(x) = 8x^3 - 15x^2 + 6x$

Q: Can I use the power rule to find the derivative of a function with a trigonometric term?

A: No, you cannot use the power rule to find the derivative of a function with a trigonometric term. The power rule is only applicable to polynomial functions. If you have a function with a trigonometric term, you will need to use a different method to find the derivative.

Conclusion

In this article, we answered some frequently asked questions about finding the derivative of a polynomial function using the power rule. We hope that this article has been helpful in clarifying any confusion you may have had about the power rule and its application to polynomial functions.

Practice Problems

1. Find the derivative of the function $f(x) = 2x^3 - 3x^2 + 1$.
2. Find the derivative of the function $f(x) = x^4 - 2x^3 + 3x^2$.
3. Find the derivative of the function $f(x) = 4x^2 - 5x + 2$.

Solutions

1. $f'(x) = 6x^2 - 6x$
2. $f'(x) = 4x^3 - 6x^2 + 6x$
3. $f'(x) = 8x - 5$

Discussion

The power rule is a powerful tool for finding derivatives, and it can be applied to any term in a polynomial function. However, it is essential to remember that the power rule is only applicable to polynomial functions and not to functions with trigonometric terms. If you have a function with a trigonometric term, you will need to use a different method to find the derivative.