Differentiate The Following Function:${ F(x) = E^{15x} }$Find { F^{\prime}(x) $}$.

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Introduction

In this article, we will focus on differentiating the given function $f(x) = e^{15x}$ and finding its derivative $f^{\prime}(x)$. The function in question is an exponential function, and we will use the properties of exponential functions to find its derivative.

What is Differentiation?

Differentiation is a fundamental concept in calculus that involves finding the rate of change of a function with respect to one of its variables. It is a powerful tool used to analyze and model various phenomena in mathematics, physics, engineering, and other fields.

The Exponential Function

The exponential function $f(x) = e^{ax}$, where $a$ is a constant, is a fundamental function in mathematics. It is defined as the inverse of the natural logarithm function, and it has several important properties. One of the key properties of the exponential function is that its derivative is equal to the function itself multiplied by the constant $a$.

Finding the Derivative of $f(x) = e^{15x}$

To find the derivative of $f(x) = e^{15x}$, we will use the property of the exponential function mentioned above. We know that the derivative of $f(x) = e^{ax}$ is equal to $f^{\prime}(x) = ae^{ax}$. In this case, $a = 15$, so we can write:

$$f^{\prime}(x) = 15e^{15x}$$

Properties of the Derivative

The derivative of $f(x) = e^{15x}$ has several important properties. One of the key properties is that it is also an exponential function. This means that the derivative of $f(x) = e^{15x}$ is also an exponential function, and it can be written in the form $f^{\prime}(x) = Ae^{Bx}$, where $A$ and $B$ are constants.

Applications of Differentiation

Differentiation has numerous applications in various fields, including physics, engineering, economics, and computer science. Some of the key applications of differentiation include:

• Finding the rate of change of a function with respect to one of its variables
• Finding the maximum or minimum value of a function
• Finding the slope of a tangent line to a curve
• Modeling population growth and decay
• Modeling the motion of objects under the influence of gravity or other forces

Conclusion

In this article, we have focused on differentiating the given function $f(x) = e^{15x}$ and finding its derivative $f^{\prime}(x)$. We have used the properties of exponential functions to find the derivative, and we have discussed the importance of differentiation in various fields. We hope that this article has provided a clear and concise explanation of the concept of differentiation and its applications.

Final Answer

The final answer to the problem is:

$$f^{\prime}(x) = 15e^{15x}$$

References

• [1] Calculus, 3rd edition, by Michael Spivak
• [2] Calculus, 2nd edition, by James Stewart
• [3] Exponential Functions, by Wolfram MathWorld

Additional Resources

• [1] Khan Academy: Differentiation
• [2] MIT OpenCourseWare: Calculus
• [3] Wolfram Alpha: Exponential Functions
  

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Introduction

In our previous article, we focused on differentiating the given function $f(x) = e^{15x}$ and finding its derivative $f^{\prime}(x)$. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q&A

Q1: What is the derivative of $f(x) = e^{15x}$?

A1: The derivative of $f(x) = e^{15x}$ is $f^{\prime}(x) = 15e^{15x}$.

Q2: Why is the derivative of $f(x) = e^{15x}$ also an exponential function?

A2: The derivative of $f(x) = e^{15x}$ is also an exponential function because the derivative of an exponential function is another exponential function. In this case, the derivative is $f^{\prime}(x) = 15e^{15x}$, which is also an exponential function.

Q3: What are some of the key applications of differentiation?

A3: Some of the key applications of differentiation include:

• Finding the rate of change of a function with respect to one of its variables
• Finding the maximum or minimum value of a function
• Finding the slope of a tangent line to a curve
• Modeling population growth and decay
• Modeling the motion of objects under the influence of gravity or other forces

Q4: How do I find the derivative of a function using the chain rule?

A4: To find the derivative of a function using the chain rule, you need to follow these steps:

1. Identify the outer function and the inner function.
2. Find the derivative of the outer function.
3. Find the derivative of the inner function.
4. Multiply the derivative of the outer function by the derivative of the inner function.

Q5: What is the difference between the derivative and the integral of a function?

A5: The derivative of a function is the rate of change of the function with respect to one of its variables, while the integral of a function is the accumulation of the function over a given interval.

Q6: How do I use the fundamental theorem of calculus to find the definite integral of a function?

A6: To use the fundamental theorem of calculus to find the definite integral of a function, you need to follow these steps:

1. Find the antiderivative of the function.
2. Evaluate the antiderivative at the upper and lower limits of integration.
3. Subtract the value of the antiderivative at the lower limit of integration from the value of the antiderivative at the upper limit of integration.

Q7: What are some of the key properties of the derivative of an exponential function?

A7: Some of the key properties of the derivative of an exponential function include:

• The derivative of an exponential function is another exponential function.
• The derivative of an exponential function is equal to the function itself multiplied by the constant $a$.
• The derivative of an exponential function is always positive.

Q8: How do I use the derivative to find the maximum or minimum value of a function?

A8: To use the derivative to find the maximum or minimum value of a function, you need to follow these steps:

1. Find the derivative of the function.
2. Set the derivative equal to zero and solve for the variable.
3. Evaluate the function at the critical points to determine the maximum or minimum value.

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts or questions that readers may have about differentiating the given function $f(x) = e^{15x}$ and finding its derivative $f^{\prime}(x)$. We hope that this article has provided a clear and concise explanation of the concept of differentiation and its applications.

Final Answer

The final answer to the problem is:

$$f^{\prime}(x) = 15e^{15x}$$

References

• [1] Calculus, 3rd edition, by Michael Spivak
• [2] Calculus, 2nd edition, by James Stewart
• [3] Exponential Functions, by Wolfram MathWorld

Additional Resources

• [1] Khan Academy: Differentiation
• [2] MIT OpenCourseWare: Calculus
• [3] Wolfram Alpha: Exponential Functions