Problem 14
Question
Calculate the derivative with respect to \(x\) of the given expression. \(\ln (-3 x)\)
Step-by-Step Solution
Verified Answer
The derivative of \(\ln(-3x)\) with respect to \(x\) is \(\frac{1}{x}\).
1Step 1: Understanding the Natural Logarithm
The expression involves the natural logarithm function \(c ln(z)\). The derivative of \(\ln(z)\) with respect to \(z\) is \(\frac{1}{z}\). We need to apply this knowledge while considering the function inside the logarithm, \(-3x\).
2Step 2: Applying the Chain Rule
The function inside the logarithm is \(-3x\), so we need to use the chain rule. The chain rule states that the derivative of a composite function \(\ln(-3x)\) is the derivative of the outer function \(\ln(z)\), \(\frac{1}{z}\), evaluated at \(z = -3x\), times the derivative of the inner function \(z = -3x\).
3Step 3: Finding the Derivative of the Inner Function
The inner function is \(-3x\), so we take the derivative with respect to \(x\). The derivative of \(-3x\) with respect to \(x\) is \(-3\).
4Step 4: Combining the Results
Use the chain rule: Multiply the derivative of the outer function \(\frac{1}{-3x}\) by the derivative of the inner function \(-3\). This gives \(\frac{1}{-3x} \times (-3)\).
5Step 5: Simplifying the Expression
Simplify \(\frac{1}{-3x} \times (-3)\) to get \(\frac{-3}{-3x} = \frac{1}{x}\). This is the final derivative of \(\ln(-3x)\) with respect to \(x\).
Key Concepts
Natural LogarithmChain RuleComposite FunctionDerivative of Inner Function
Natural Logarithm
The natural logarithm, often represented as \( \ln(x) \), is a fundamental concept in calculus involving logarithms, where the base is the constant \( e \), approximately equal to 2.71828. The natural logarithm function is used to solve problems related to growth and decay, and it is also frequently encountered in mathematical derivatives.
The derivative of the natural logarithm function \( \ln(x) \) is \( \frac{1}{x} \). This property is crucial when dealing with expressions inside the logarithm, as it helps us to simplify and manipulate the function effectively.
Understanding how the natural logarithm works is the foundational step toward tackling more complex derivatives involving logs and other functions.
The derivative of the natural logarithm function \( \ln(x) \) is \( \frac{1}{x} \). This property is crucial when dealing with expressions inside the logarithm, as it helps us to simplify and manipulate the function effectively.
Understanding how the natural logarithm works is the foundational step toward tackling more complex derivatives involving logs and other functions.
Chain Rule
The chain rule in calculus is a technique used to differentiate composite functions. It is essential because not all functions are simple; many require a method to handle the "chain" of functions involved in their composition.
The chain rule states that the derivative of a function \( f(g(x)) \) is given by \( f'(g(x)) \times g'(x) \). In simple terms, it means you take the derivative of the outer function and multiply it by the derivative of the inner function.
This rule helps you break down a problem into manageable parts, allowing you to apply basic derivative rules to individual components. For example, in the expression \( \ln(-3x) \), the chain rule helps determine the derivative by differentiating the outer function \( \ln(z) \) and the inner function \( z = -3x \), and then combining the results.
The chain rule states that the derivative of a function \( f(g(x)) \) is given by \( f'(g(x)) \times g'(x) \). In simple terms, it means you take the derivative of the outer function and multiply it by the derivative of the inner function.
This rule helps you break down a problem into manageable parts, allowing you to apply basic derivative rules to individual components. For example, in the expression \( \ln(-3x) \), the chain rule helps determine the derivative by differentiating the outer function \( \ln(z) \) and the inner function \( z = -3x \), and then combining the results.
Composite Function
Composite functions occur when one function is applied to the result of another function. They can be recognized in expressions like \( \ln(-3x) \) where \( f(x) = \ln(x) \) and \( g(x) = -3x \). The notation for a composite function is usually \( f(g(x)) \), which indicates \( g(x) \) is computed first, and then \( f \) is applied to this result.
When dealing with derivatives of composite functions, it is crucial to identify the inner and outer functions. Recognizing this structure allows for the effective application of the chain rule. In our example, understanding that \( \ln(-3x) \) is a composite function helps simplify finding the derivative by focusing on \( \ln(z) \) and \(-3x\) separately, before applying the chain rule to combine their derivatives.
When dealing with derivatives of composite functions, it is crucial to identify the inner and outer functions. Recognizing this structure allows for the effective application of the chain rule. In our example, understanding that \( \ln(-3x) \) is a composite function helps simplify finding the derivative by focusing on \( \ln(z) \) and \(-3x\) separately, before applying the chain rule to combine their derivatives.
Derivative of Inner Function
The inner function in a composite structure refers to the function inside another function, such as \(-3x\) in \( \ln(-3x) \). Differentiating this inner function is an important step when using the chain rule.
For the inner function \(-3x\), the derivative with respect to \(x\) is a straightforward calculation, yielding \(-3\).
This derivative becomes a key part in calculating the overall derivative of the composite function. It must be multiplied with the derivative of the outer function (evaluated at the inner function) to arrive at the complete derivative. This step exemplifies the application of the chain rule, where each part of a composite function is tackled individually, then combined for the final result.
For the inner function \(-3x\), the derivative with respect to \(x\) is a straightforward calculation, yielding \(-3\).
This derivative becomes a key part in calculating the overall derivative of the composite function. It must be multiplied with the derivative of the outer function (evaluated at the inner function) to arrive at the complete derivative. This step exemplifies the application of the chain rule, where each part of a composite function is tackled individually, then combined for the final result.
Other exercises in this chapter
Problem 14
Calculate the lower and upper Riemann sums for the given function \(f\), interval \(I\), and uniform partition of order 2 $$ f(x)=x^{2}-2 x+2 \quad I=[0,3] $$
View solution Problem 14
Suppose that \(\int_{6}^{8}(3 f(x)-x) d x=6\) and \(\int_{8}^{6}(2 x+4 g(x)) d x=-8 .\) Evaluate \(\int_{8}^{6}(f(x)-5 g(x)) d x\).
View solution Problem 14
In Exercises \(11-20,\) calculate \(F(x)=\int_{a}^{x} f(t) d t\) $$= f(t)=e^{-t} \quad a=-1 $$
View solution Problem 14
In Exercises \(11-16,\) use summation notation to express the sum. $$ 9+16+25+36 $$
View solution