Determine Whether The Statement Is True Or False.If F ( X ) = Π X F(x)=\pi^x F ( X ) = Π X , Then F ′ ( X ) = X Π ( X − 1 ) F^{\prime}(x)=x \pi^{(x-1)} F ′ ( X ) = X Π ( X − 1 ) .A. True B. False

Introduction

In mathematics, the concept of derivatives plays a crucial role in understanding the behavior of functions. 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 examine the validity of a given statement regarding the derivative of the function $f(x) = \pi^x$. The statement claims that the derivative of $f(x)$ is $f^{\prime}(x) = x \pi^{(x-1)}$. Our objective is to determine whether this statement is true or false.

Understanding the Function and Its Derivative

The given function is $f(x) = \pi^x$. To find the derivative of this function, we can use the chain rule, which states that if $f(x) = g(h(x))$, then $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$. In this case, we can rewrite the function as $f(x) = e^{x \ln \pi}$, where $\ln \pi$ is the natural logarithm of $\pi$.

Applying the Chain Rule

To find the derivative of $f(x)$, we can apply the chain rule. Let $g(x) = e^x$ and $h(x) = x \ln \pi$. Then, we have:

$$f(x) = g(h(x)) = e^{x \ln \pi}$$

Using the chain rule, we get:

$$f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x) = e^{x \ln \pi} \cdot \ln \pi$$

Simplifying the expression, we get:

$$f^{\prime}(x) = \pi^x \ln \pi$$

Comparing the Derivative with the Given Statement

Now, let's compare the derivative we obtained with the given statement. The statement claims that the derivative of $f(x)$ is $f^{\prime}(x) = x \pi^{(x-1)}$. However, our calculation shows that the derivative is actually $f^{\prime}(x) = \pi^x \ln \pi$. These two expressions are not equal, which means that the given statement is false.

Conclusion

In conclusion, the statement that the derivative of $f(x) = \pi^x$ is $f^{\prime}(x) = x \pi^{(x-1)}$ is false. The correct derivative of the function is $f^{\prime}(x) = \pi^x \ln \pi$. This result highlights the importance of carefully applying mathematical rules and formulas to obtain accurate results.

Final Answer

The final answer is: B. False

Additional Information

For those who are interested in learning more about derivatives and their applications, here are some additional resources:

• Derivative Rules: A comprehensive guide to the rules of differentiation, including the power rule, product rule, and quotient rule.
• Derivative Applications: A collection of examples and exercises that demonstrate the practical applications of derivatives in various fields, such as physics, engineering, and economics.
• Mathematical Functions: A detailed overview of common mathematical functions, including exponential, logarithmic, and trigonometric functions.

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 one of its variables. It is a measure of how fast the function changes as its input changes.

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

A: There are several rules for finding the derivative of a function, including the power rule, product rule, and quotient rule. These rules can be applied to various types of functions, such as polynomial, exponential, and trigonometric functions.

Q: What is the power rule of differentiation?

A: The power rule of differentiation states that if $f(x) = x^n$, then $f^{\prime}(x) = nx^{n-1}$. This rule can be applied to any function of the form $f(x) = x^n$, where $n$ is a constant.

Q: What is the product rule of differentiation?

A: The product rule of differentiation states that if $f(x) = u(x)v(x)$, then $f^{\prime}(x) = u^{\prime}(x)v(x) + u(x)v^{\prime}(x)$. This rule can be applied to any function of the form $f(x) = u(x)v(x)$, where $u(x)$ and $v(x)$ are both functions of $x$.

Q: What is the quotient rule of differentiation?

A: The quotient rule of differentiation states that if $f(x) = \frac{u(x)}{v(x)}$, then $f^{\prime}(x) = \frac{u^{\prime}(x)v(x) - u(x)v^{\prime}(x)}{[v(x)]^2}$. This rule can be applied to any function of the form $f(x) = \frac{u(x)}{v(x)}$, where $u(x)$ and $v(x)$ are both functions of $x$.

Q: What is the chain rule of differentiation?

A: The chain rule of differentiation states that if $f(x) = g(h(x))$, then $f^{\prime}(x) = g^{\prime}(h(x)) \cdot h^{\prime}(x)$. This rule can be applied to any function of the form $f(x) = g(h(x))$, where $g(x)$ and $h(x)$ are both functions of $x$.

Q: What is the derivative of the function $f(x) = \pi^x$?

A: The derivative of the function $f(x) = \pi^x$ is $f^{\prime}(x) = \pi^x \ln \pi$. This result can be obtained by applying the chain rule and the fact that the derivative of $e^x$ is $e^x$.

Q: What is the derivative of the function $f(x) = \sin x$?

A: The derivative of the function $f(x) = \sin x$ is $f^{\prime}(x) = \cos x$. This result can be obtained by applying the fact that the derivative of $\sin x$ is $\cos x$.

Q: What is the derivative of the function $f(x) = \cos x$?

A: The derivative of the function $f(x) = \cos x$ is $f^{\prime}(x) = -\sin x$. This result can be obtained by applying the fact that the derivative of $\cos x$ is $-\sin x$.

Q: What is the derivative of the function $f(x) = e^x$?

A: The derivative of the function $f(x) = e^x$ is $f^{\prime}(x) = e^x$. This result can be obtained by applying the fact that the derivative of $e^x$ is $e^x$.

Q: What is the derivative of the function $f(x) = \ln x$?

A: The derivative of the function $f(x) = \ln x$ is $f^{\prime}(x) = \frac{1}{x}$. This result can be obtained by applying the fact that the derivative of $\ln x$ is $\frac{1}{x}$.

Conclusion

In conclusion, the derivative of a function represents the rate of change of the function with respect to one of its variables. There are several rules for finding the derivative of a function, including the power rule, product rule, quotient rule, and chain rule. By applying these rules, we can find the derivative of various types of functions, including polynomial, exponential, and trigonometric functions.