Find The Derivative Of The Given Function.Given: \[$ F(x) = 6x^5 - 2x^4 + X^3 - 5x + 7 \$\]Calculate: \[$ F'(x) = \$\] $\[ \square \\]

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 given polynomial function. We will use the power rule of differentiation, which is a fundamental concept in calculus.

The Power Rule of Differentiation

The power rule of differentiation states that if we have a function of the form:

$$f(x) = x^n$$

where $n$ is a constant, then the derivative of the function is:

$$f'(x) = nx^{n-1}$$

This rule can be extended to more complex functions by applying it to each term separately.

Finding the Derivative of the Given Function

The given function is:

$$f(x) = 6x^5 - 2x^4 + x^3 - 5x + 7$$

To find the derivative of this function, we will apply the power rule of differentiation to each term separately.

Term 1: $6x^5$

Using the power rule, we have:

$$\frac{d}{dx}(6x^5) = 6 \cdot 5x^{5-1} = 30x^4$$

Term 2: $-2x^4$

Using the power rule, we have:

$$\frac{d}{dx}(-2x^4) = -2 \cdot 4x^{4-1} = -8x^3$$

Term 3: $x^3$

Using the power rule, we have:

$$\frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2$$

Term 4: $-5x$

Using the power rule, we have:

$$\frac{d}{dx}(-5x) = -5 \cdot 1x^{1-1} = -5$$

Term 5: $7$

The derivative of a constant is zero, so:

$$\frac{d}{dx}(7) = 0$$

Combining the Derivatives

Now that we have found the derivative of each term, we can combine them to find the derivative of the given function:

$$f'(x) = 30x^4 - 8x^3 + 3x^2 - 5$$

Conclusion

In this article, we have used the power rule of differentiation to find the derivative of a given polynomial function. We have applied the power rule to each term separately and then combined the derivatives to find the final result. The derivative of the given function is:

$$f'(x) = 30x^4 - 8x^3 + 3x^2 - 5$$

This result can be used to find the rate of change of the function with respect to its input.

Example Use Cases

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

• Optimization: The derivative of a function can be used to find the maximum or minimum value of the function.
• Physics: The derivative of a function can be used to model the motion of an object.
• Economics: The derivative of a function can be used to model the behavior of a market.

Common Mistakes to Avoid

When finding the derivative of a function, there are several common mistakes to avoid, including:

• Forgetting to apply the power rule: Make sure to apply the power rule to each term separately.
• Making errors in the exponent: Double-check the exponent of each term to ensure that it is correct.
• Forgetting to combine the derivatives: Make sure to combine the derivatives of each term to find the final result.

Conclusion

In conclusion, finding the derivative of a function is a fundamental concept in calculus. By applying the power rule of differentiation, we can find the derivative of a given polynomial function. The derivative of the given function is:

$$f'(x) = 30x^4 - 8x^3 + 3x^2 - 5$$

Introduction

In our previous article, we discussed how to find the derivative of a given polynomial function using the power rule of differentiation. In this article, we will provide a Q&A guide to help you understand the concept of derivatives and how to apply the power rule.

Q: What is the derivative of a function?

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

Q: What is the power rule of differentiation?

A: The power rule of differentiation states that if we have a function of the form:

$$f(x) = x^n$$

where $n$ is a constant, then the derivative of the function is:

$$f'(x) = nx^{n-1}$$

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 apply it to each term separately. For example, if we have the function:

$$f(x) = 6x^5 - 2x^4 + x^3 - 5x + 7$$

We would apply the power rule to each term separately:

• For the term $6x^5$, we would have: $\frac{d}{dx}(6x^5) = 6 \cdot 5x^{5-1} = 30x^4$
• For the term $-2x^4$, we would have: $\frac{d}{dx}(-2x^4) = -2 \cdot 4x^{4-1} = -8x^3$
• For the term $x^3$, we would have: $\frac{d}{dx}(x^3) = 3x^{3-1} = 3x^2$
• For the term $-5x$, we would have: $\frac{d}{dx}(-5x) = -5 \cdot 1x^{1-1} = -5$
• For the term $7$, we would have: $\frac{d}{dx}(7) = 0$

Q: What is the derivative of a constant?

A: The derivative of a constant is zero. This is because a constant does not change as its input changes.

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 we have the function:

$$f(x) = x^{-2}$$

We would apply the power rule to get:

$$f'(x) = -2x^{-2-1} = -2x^{-3}$$

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 we have the function:

$$f(x) = x^{1/2}$$

We would apply the power rule to get:

$$f'(x) = \frac{1}{2}x^{1/2-1} = \frac{1}{2}x^{-1/2}$$

Q: What are some common mistakes to avoid when finding the derivative of a function?

A: Some common mistakes to avoid when finding the derivative of a function include:

• Forgetting to apply the power rule
• Making errors in the exponent
• Forgetting to combine the derivatives
• Not checking the domain of the function

Conclusion

In conclusion, finding the derivative of a function is a fundamental concept in calculus. By applying the power rule of differentiation, we can find the derivative of a given polynomial function. We hope that this Q&A guide has helped you understand the concept of derivatives and how to apply the power rule.