If $f(x)=\frac{7 X}{\ln X}$, Find $f^{\prime}(e$\].$f^{\prime}(e) = $

If $f(x)=\frac{7 x}{\ln x}$, find $f^{\prime}(e)$

In this article, we will explore the concept of finding the derivative of a given function, specifically $f(x)=\frac{7 x}{\ln x}$, at a specific point, $e$. The derivative of a function represents the rate of change of the function with respect to its input. In this case, we are interested in finding the derivative of $f(x)$ at $x=e$, where $e$ is a mathematical constant approximately equal to 2.71828.

Before we dive into finding the derivative, let's take a closer look at the function $f(x)=\frac{7 x}{\ln x}$. This function is a quotient of two functions: $7x$ and $\ln x$. The natural logarithm function, denoted by $\ln x$, is the inverse of the exponential function. In other words, $\ln x$ is the power to which the base $e$ must be raised to produce the number $x$.

To find the derivative of $f(x)$, we will use the quotient rule, which states that if $f(x)=\frac{g(x)}{h(x)}$, then $f^{\prime}(x)=\frac{h(x)g^{\prime}(x)-g(x)h^{\prime}(x)}{(h(x))^2}$. In this case, $g(x)=7x$ and $h(x)=\ln x$.

Using the quotient rule, we can find the derivative of $f(x)$ as follows:

$$f^{\prime}(x)=\frac{(\ln x)(7)-7(\ln x)^{\prime}}{(\ln x)^2}$$

Now, we need to find the derivative of $\ln x$, which is $\frac{1}{x}$. Substituting this into the previous equation, we get:

$$f^{\prime}(x)=\frac{7\ln x-7\left(\frac{1}{x}\right)}{(\ln x)^2}$$

Simplifying the expression, we get:

$$f^{\prime}(x)=\frac{7\ln x-\frac{7}{x}}{(\ln x)^2}$$

Now that we have found the derivative of $f(x)$, we can evaluate it at $x=e$. Substituting $x=e$ into the previous equation, we get:

$$f^{\prime}(e)=\frac{7\ln e-\frac{7}{e}}{(\ln e)^2}$$

Since $\ln e=1$, we can simplify the expression as follows:

$$f^{\prime}(e)=\frac{7-\frac{7}{e}}{1}$$

Simplifying further, we get:

$$f^{\prime}(e)=7-\frac{7}{e}$$

In this article, we found the derivative of the function $f(x)=\frac{7 x}{\ln x}$ using the quotient rule. We then evaluated the derivative at $x=e$ to find the value of $f^{\prime}(e)$. The final answer is:

$f^{\prime}(e) = 7-\frac{7}{e}$

This result provides valuable insight into the behavior of the function $f(x)$ at the point $x=e$.
Q&A: If $f(x)=\frac{7 x}{\ln x}$, find $f^{\prime}(e)$

In our previous article, we explored the concept of finding the derivative of a given function, specifically $f(x)=\frac{7 x}{\ln x}$, at a specific point, $e$. We found that the derivative of $f(x)$ is given by $f^{\prime}(x)=\frac{7\ln x-\frac{7}{x}}{(\ln x)^2}$ and evaluated it at $x=e$ to find the value of $f^{\prime}(e)$. In this article, we will answer some common questions related to this topic.

A: The quotient rule is a formula for finding the derivative of a quotient of two functions. If $f(x)=\frac{g(x)}{h(x)}$, then $f^{\prime}(x)=\frac{h(x)g^{\prime}(x)-g(x)h^{\prime}(x)}{(h(x))^2}$.

A: To apply the quotient rule, we need to identify the two functions $g(x)$ and $h(x)$ in the quotient. In this case, $g(x)=7x$ and $h(x)=\ln x$. We then find the derivatives of $g(x)$ and $h(x)$, which are $g^{\prime}(x)=7$ and $h^{\prime}(x)=\frac{1}{x}$. Finally, we substitute these values into the quotient rule formula to find the derivative of $f(x)$.

A: We found that the derivative of $f(x)$ is given by $f^{\prime}(x)=\frac{7\ln x-\frac{7}{x}}{(\ln x)^2}$. Evaluating this at $x=e$, we get $f^{\prime}(e)=7-\frac{7}{e}$.

A: The derivative of $f(x)$ represents the rate of change of the function with respect to its input. In this case, we are interested in finding the derivative of $f(x)$ at $x=e$ to understand the behavior of the function at this point.

A: Yes, the quotient rule can be used to find the derivative of any quotient of two functions. However, you need to identify the two functions $g(x)$ and $h(x)$ in the quotient and find their derivatives.

A: The quotient rule has many applications in calculus, including finding the derivative of rational functions, trigonometric functions, and exponential functions. It is also used in physics and engineering to model real-world phenomena.

In this article, we answered some common questions related to finding the derivative of a given function, specifically $f(x)=\frac{7 x}{\ln x}$, at a specific point, $e$. We hope that this article has provided valuable insight into the concept of the quotient rule and its applications.