Find D D X ( 5 Ln ⁡ ( 2 X + 7 ) \frac{d}{d X}(5 \ln (2 X+7) D X D ​ ( 5 Ln ( 2 X + 7 ) ].

Introduction

In calculus, 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 focus on finding the derivative of a natural logarithm function, specifically the function $\frac{d}{d x}(5 \ln (2 x+7))$. This type of function is commonly encountered in various fields, including physics, engineering, and economics.

The Natural Logarithm Function

The natural logarithm function, denoted by $\ln x$, is a fundamental function in mathematics that has numerous applications in various fields. It is defined as the inverse of the exponential function $e^x$, where $e$ is a mathematical constant approximately equal to 2.71828. The natural logarithm function has several important properties, including:

• Domain: The domain of the natural logarithm function is all positive real numbers, i.e., $x > 0$.
• Range: The range of the natural logarithm function is all real numbers, i.e., $-\infty < \ln x < \infty$.
• Derivative: The derivative of the natural logarithm function is $\frac{1}{x}$.

Applying the Chain Rule

To find the derivative of the function $\frac{d}{d x}(5 \ln (2 x+7))$, we will apply the chain rule, which is a fundamental rule in calculus for differentiating composite functions. The chain rule states that if we have a composite function of the form $f(g(x))$, then the derivative of the composite function is given by:

$$\frac{d}{d x} f(g(x)) = f'(g(x)) \cdot g'(x)$$

In our case, we have a composite function of the form $5 \ln (2 x+7)$, where the outer function is the natural logarithm function and the inner function is $2 x+7$. We will apply the chain rule to find the derivative of this composite function.

Finding the Derivative

Using the chain rule, we can write the derivative of the composite function as:

$$\frac{d}{d x} (5 \ln (2 x+7)) = 5 \cdot \frac{1}{2 x+7} \cdot \frac{d}{d x} (2 x+7)$$

Now, we need to find the derivative of the inner function $2 x+7$. Using the power rule, we can write:

$$\frac{d}{d x} (2 x+7) = 2$$

Substituting this result back into the previous equation, we get:

$$\frac{d}{d x} (5 \ln (2 x+7)) = 5 \cdot \frac{1}{2 x+7} \cdot 2$$

Simplifying this expression, we get:

$$\frac{d}{d x} (5 \ln (2 x+7)) = \frac{10}{2 x+7}$$

Conclusion

In this article, we have found the derivative of the natural logarithm function $\frac{d}{d x}(5 \ln (2 x+7))$ using the chain rule. The derivative of this function is $\frac{10}{2 x+7}$. This result has numerous applications in various fields, including physics, engineering, and economics.

Example Problems

Here are some example problems that illustrate the application of the chain rule in finding the derivative of a natural logarithm function:

Example 1

Find the derivative of the function $\frac{d}{d x}(3 \ln (x^2+1))$.

Solution

Using the chain rule, we can write the derivative of the composite function as:

$$\frac{d}{d x} (3 \ln (x^2+1)) = 3 \cdot \frac{1}{x^2+1} \cdot \frac{d}{d x} (x^2+1)$$

Now, we need to find the derivative of the inner function $x^2+1$. Using the power rule, we can write:

$$\frac{d}{d x} (x^2+1) = 2 x$$

Substituting this result back into the previous equation, we get:

$$\frac{d}{d x} (3 \ln (x^2+1)) = 3 \cdot \frac{1}{x^2+1} \cdot 2 x$$

Simplifying this expression, we get:

$$\frac{d}{d x} (3 \ln (x^2+1)) = \frac{6 x}{x^2+1}$$

Example 2

Find the derivative of the function $\frac{d}{d x}(4 \ln (2 x-3))$.

Solution

Using the chain rule, we can write the derivative of the composite function as:

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

Now, we need to find the derivative of the inner function $2 x-3$. Using the power rule, we can write:

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

Substituting this result back into the previous equation, we get:

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

Simplifying this expression, we get:

$$\frac{d}{d x} (4 \ln (2 x-3)) = \frac{8}{2 x-3}$$

Final Answer

$$

Introduction

In our previous article, we discussed how to find the derivative of a natural logarithm function using the chain rule. In this article, we will provide a Q&A section to help clarify any doubts and provide additional examples.

Q: What is the chain rule?

A: The chain rule is a fundamental rule in calculus for differentiating composite functions. It states that if we have a composite function of the form $f(g(x))$, then the derivative of the composite function is given by:

$$\frac{d}{d x} f(g(x)) = f'(g(x)) \cdot g'(x)$$

Q: How do I apply the chain rule to find the derivative of a natural logarithm function?

A: To apply the chain rule to find the derivative of a natural logarithm function, 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. Substitute the derivative of the inner function into the derivative of the outer function.
5. Simplify the expression.

Q: What is the derivative of the natural logarithm function?

A: The derivative of the natural logarithm function is $\frac{1}{x}$.

Q: How do I find the derivative of a composite function with a natural logarithm function as the outer function?

A: To find the derivative of a composite function with a natural logarithm function as the outer function, follow these steps:

1. Identify the inner function.
2. Find the derivative of the inner function.
3. Substitute the derivative of the inner function into the derivative of the natural logarithm function.
4. Simplify the expression.

Q: What is the derivative of the function $\frac{d}{d x}(3 \ln (x^2+1))$?

A: Using the chain rule, we can write the derivative of the composite function as:

$$\frac{d}{d x} (3 \ln (x^2+1)) = 3 \cdot \frac{1}{x^2+1} \cdot \frac{d}{d x} (x^2+1)$$

Now, we need to find the derivative of the inner function $x^2+1$. Using the power rule, we can write:

$$\frac{d}{d x} (x^2+1) = 2 x$$

Substituting this result back into the previous equation, we get:

$$\frac{d}{d x} (3 \ln (x^2+1)) = 3 \cdot \frac{1}{x^2+1} \cdot 2 x$$

Simplifying this expression, we get:

$$\frac{d}{d x} (3 \ln (x^2+1)) = \frac{6 x}{x^2+1}$$

Q: What is the derivative of the function $\frac{d}{d x}(4 \ln (2 x-3))$?

A: Using the chain rule, we can write the derivative of the composite function as:

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

Now, we need to find the derivative of the inner function $2 x-3$. Using the power rule, we can write:

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

Substituting this result back into the previous equation, we get:

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

Simplifying this expression, we get:

$$\frac{d}{d x} (4 \ln (2 x-3)) = \frac{8}{2 x-3}$$

Conclusion

In this article, we have provided a Q&A section to help clarify any doubts and provide additional examples on finding the derivative of a natural logarithm function using the chain rule. We hope this article has been helpful in understanding the concept of the chain rule and its application to finding the derivative of a natural logarithm function.

Final Answer

The final answer is $\boxed{\frac{10}{2 x+7}}$.