Find $f^ \prime}(x)$ For The Following Function. Then Find $f^{\prime}(2)$, $ F ′ ( 0 ) F^{\prime}(0) F ′ ( 0 ) [/tex], And $f^{\prime}(-3)$.Given $f(x) = 6x^2 + 5x$$f^{\prime (x) =$[/tex]

Introduction

In calculus, the derivative of a function represents the rate of change of the function with respect to its input. It is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the derivative of a given function and then evaluating the derivative at specific points.

The Power Rule

The power rule is a fundamental rule in calculus that states if we have a function of the form $f(x) = x^n$, then the derivative of the function is given by $f^{\prime}(x) = nx^{n-1}$. This rule can be extended to functions of the form $f(x) = ax^n$, where $a$ is a constant.

Finding the Derivative of the Given Function

Given the function $f(x) = 6x^2 + 5x$, we can use the power rule to find the derivative of the function. We will first find the derivative of the term $6x^2$ and then the derivative of the term $5x$.

The derivative of $6x^2$ is given by $f^{\prime}(x) = 12x$. This is because the power rule states that if we have a function of the form $f(x) = x^n$, then the derivative of the function is given by $f^{\prime}(x) = nx^{n-1}$. In this case, $n = 2$, so the derivative of $6x^2$ is $12x$.

The derivative of $5x$ is given by $f^{\prime}(x) = 5$. This is because the power rule states that if we have a function of the form $f(x) = ax^n$, then the derivative of the function is given by $f^{\prime}(x) = anx^{n-1}$. In this case, $a = 5$ and $n = 1$, so the derivative of $5x$ is $5$.

Combining the Derivatives

Now that we have found the derivatives of the individual terms, we can combine them to find the derivative of the entire function. The derivative of the function $f(x) = 6x^2 + 5x$ is given by $f^{\prime}(x) = 12x + 5$.

Evaluating the Derivative at Specific Points

Now that we have found the derivative of the function, we can evaluate the derivative at specific points. We will evaluate the derivative at $x = 2$, $x = 0$, and $x = -3$.

Evaluating the Derivative at $x = 2$

To evaluate the derivative at $x = 2$, we substitute $x = 2$ into the derivative of the function. This gives us $f^{\prime}(2) = 12(2) + 5 = 24 + 5 = 29$.

Evaluating the Derivative at $x = 0$

To evaluate the derivative at $x = 0$, we substitute $x = 0$ into the derivative of the function. This gives us $f^{\prime}(0) = 12(0) + 5 = 0 + 5 = 5$.

Evaluating the Derivative at $x = -3$

To evaluate the derivative at $x = -3$, we substitute $x = -3$ into the derivative of the function. This gives us $f^{\prime}(-3) = 12(-3) + 5 = -36 + 5 = -31$.

Conclusion

In this article, we found the derivative of the function $f(x) = 6x^2 + 5x$ using the power rule. We then evaluated the derivative at specific points, including $x = 2$, $x = 0$, and $x = -3$. The derivative of the function is given by $f^{\prime}(x) = 12x + 5$, and the values of the derivative at the specified points are $f^{\prime}(2) = 29$, $f^{\prime}(0) = 5$, and $f^{\prime}(-3) = -31$.

References

• [1] Calculus, 3rd edition, Michael Spivak
• [2] Calculus, 2nd edition, James Stewart

Glossary

• Derivative: The rate of change of a function with respect to its input.
• Power Rule: A fundamental rule in calculus that states if we have a function of the form $f(x) = x^n$, then the derivative of the function is given by $f^{\prime}(x) = nx^{n-1}$.
• Constant Multiple Rule: A rule in calculus that states if we have a function of the form $f(x) = ax^n$, then the derivative of the function is given by $f^{\prime}(x) = anx^{n-1}$.

Related Topics

• Limits: The concept of limits is a fundamental concept in calculus that deals with the behavior of functions as the input approaches a specific value.
• Integrals: The concept of integrals is a fundamental concept in calculus that deals with the accumulation of quantities over a given interval.
• Multivariable Calculus: The concept of multivariable calculus deals with functions of multiple variables and is used to study the behavior of functions in higher dimensions.
  Derivatives Q&A =====================

Introduction

In our previous article, we discussed finding the derivative of a function and evaluating the derivative at specific points. In this article, we will answer some common questions related to derivatives.

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 the input changes.

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

A: To find the derivative of a function, you can use the power rule, which states that if we have a function of the form $f(x) = x^n$, then the derivative of the function is given by $f^{\prime}(x) = nx^{n-1}$. You can also use other rules, such as the constant multiple rule and the sum rule.

Q: What is the power rule?

A: The power rule is a fundamental rule in calculus that states if we have a function of the form $f(x) = x^n$, then the derivative of the function is given by $f^{\prime}(x) = nx^{n-1}$. This rule can be extended to functions of the form $f(x) = ax^n$, where $a$ is a constant.

Q: What is the constant multiple rule?

A: The constant multiple rule is a rule in calculus that states if we have a function of the form $f(x) = ax^n$, then the derivative of the function is given by $f^{\prime}(x) = anx^{n-1}$. This rule can be used to find the derivative of a function that is a constant multiple of another function.

Q: What is the sum rule?

A: The sum rule is a rule in calculus that states if we have two functions $f(x)$ and $g(x)$, then the derivative of the sum of the two functions is given by $(f+g)^{\prime}(x) = f^{\prime}(x) + g^{\prime}(x)$. This rule can be used to find the derivative of a function that is the sum of two other functions.

Q: How do I evaluate the derivative of a function at a specific point?

A: To evaluate the derivative of a function at a specific point, you can substitute the value of the input into the derivative of the function. For example, if we have a function $f(x) = 6x^2 + 5x$ and we want to evaluate the derivative at $x = 2$, we would substitute $x = 2$ into the derivative of the function, which is $f^{\prime}(x) = 12x + 5$. This gives us $f^{\prime}(2) = 12(2) + 5 = 24 + 5 = 29$.

Q: What is the physical significance of the derivative?

A: The derivative of a function has many physical significance. For example, if we have a function that represents the position of an object as a function of time, the derivative of the function represents the velocity of the object. If we have a function that represents the velocity of an object as a function of time, the derivative of the function represents the acceleration of the object.

Q: What are some common applications of derivatives?

A: Derivatives have many common applications in physics, engineering, and economics. Some examples include:

• Motion: Derivatives are used to describe the motion of objects, including their velocity and acceleration.
• Optimization: Derivatives are used to find the maximum or minimum of a function, which is useful in optimization problems.
• Economics: Derivatives are used to model the behavior of economic systems, including the behavior of prices and quantities.
• Computer Science: Derivatives are used in computer science to model the behavior of algorithms and data structures.

Conclusion

In this article, we answered some common questions related to derivatives. We discussed the power rule, the constant multiple rule, and the sum rule, and we provided examples of how to evaluate the derivative of a function at a specific point. We also discussed the physical significance of the derivative and some common applications of derivatives.