Problem 344
Question
For the following exercises, verify the derivatives and antiderivatives. $$\frac{d}{d x} \ln \left(\frac{x-a}{x+a}\right)=\frac{2 a}{\left(x^{2}-a^{2}\right)}$$
Step-by-Step Solution
Verified Answer
The derivative is correctly verified: \( \frac{2a}{x^2-a^2} \).
1Step 1: Recall the Derivative of Natural Logarithm
The derivative of a natural log function, \( \ln(u) \), is given by \( \frac{du}{dx} \cdot \frac{1}{u} \). In this exercise, \( u = \frac{x-a}{x+a} \). We will need to find the derivative of \( u \) with respect to \( x \).
2Step 2: Use the Quotient Rule for Derivatives
To find \( \frac{du}{dx} \), use the quotient rule, which states: if \( u = \frac{f}{g} \), then \( \frac{du}{dx} = \frac{f'g - fg'}{g^2} \). Let \( f = x-a \) and \( g = x+a \). Then \( f' = 1 \) and \( g' = 1 \). Compute: \( \frac{du}{dx} = \frac{(1)(x+a) - (x-a)(1)}{(x+a)^2} \).
3Step 3: Simplify the Derivative of \( u \)
Simplify \( \frac{du}{dx} = \frac{x+a - x+a}{(x+a)^2} = \frac{2a}{(x+a)^2} \).
4Step 4: Combine Results to Find Total Derivative
Substitute back into the derivative of the natural log function: \( \frac{d}{dx} \ln\left(\frac{x-a}{x+a}\right) = \frac{1}{\frac{x-a}{x+a}} \cdot \frac{2a}{(x+a)^2} \).
5Step 5: Simplify the Expression
The expression simplifies to \( \frac{2a(x+a)}{(x+a)^2(x-a)} \). Simplifying further, \( \frac{2a}{x^2-a^2} \), as \((x+a)(x-a) = x^2-a^2\).
6Step 6: Verify the Given Derivative
The simplified expression, \( \frac{2a}{x^2-a^2} \), matches the provided derivative, which confirms that the derivative is correctly calculated.
Key Concepts
Natural LogarithmQuotient RuleSimplifying ExpressionsVerifying Derivatives
Natural Logarithm
The natural logarithm is a fundamental concept in calculus, denoted by \( \ln(x) \). It is the logarithm to the base \( e \), where \( e \approx 2.71828 \). A key property of the natural log is its derivative: if you have a function \( \ln(u) \), the derivative is calculated by the formula \( \frac{d}{dx} \ln(u) = \frac{du}{dx} \cdot \frac{1}{u} \).
This formula shows the effectiveness of the chain rule, as it involves differentiating the inside function \( u \) and then dividing by \( u \) itself. In the context of the exercise, \( u \) was considered as the fraction \( \frac{x-a}{x+a} \). By finding the derivative of this fraction, you can then apply it to the natural logarithm formula and proceed with the calculations. Understanding this concept is critical when you are working with logarithmic functions in calculus.
This formula shows the effectiveness of the chain rule, as it involves differentiating the inside function \( u \) and then dividing by \( u \) itself. In the context of the exercise, \( u \) was considered as the fraction \( \frac{x-a}{x+a} \). By finding the derivative of this fraction, you can then apply it to the natural logarithm formula and proceed with the calculations. Understanding this concept is critical when you are working with logarithmic functions in calculus.
- Logarithms simplify multiplication into addition and division into subtraction.
- The natural log has unique scaling properties used in many applications.
Quotient Rule
The quotient rule is essential when differentiating a function that can be expressed as a quotient or fraction, \( \frac{f(x)}{g(x)} \). The quotient rule states that:
\[\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2}\]
Here, \( f \) and \( g \) are both functions of \( x \), with \( f' \) and \( g' \) denoting their derivatives. In the exercise problem, we used \( f = x-a \) and \( g = x+a \). The derivation begins by finding the derivatives \( f' = 1 \) and \( g' = 1 \).
This straightforward application of the quotient rule allows for the computation of the derivative of the fraction \( \frac{x-a}{x+a} \), resulting in \( \frac{2a}{(x+a)^2} \).
\[\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f'g - fg'}{g^2}\]
Here, \( f \) and \( g \) are both functions of \( x \), with \( f' \) and \( g' \) denoting their derivatives. In the exercise problem, we used \( f = x-a \) and \( g = x+a \). The derivation begins by finding the derivatives \( f' = 1 \) and \( g' = 1 \).
This straightforward application of the quotient rule allows for the computation of the derivative of the fraction \( \frac{x-a}{x+a} \), resulting in \( \frac{2a}{(x+a)^2} \).
- Remember: the denominator in the quotient formula is always squared \( (g^2) \).
- This rule is crucial when differentiating expressions involving division of variables.
Simplifying Expressions
After applying derivation rules, the next step is usually to simplify the resulting expression for easier interpretation and utilization. Simplification might involve applying algebraic techniques, such as factoring or reducing fractions.
Referring to the solution of the problem, the expression \( \frac{2a(x+a)}{(x+a)^2(x-a)} \) was simplified. By expanding \((x+a)(x-a)\) as \(x^2 - a^2\), the expression simplifies to \(\frac{2a}{x^2 - a^2}\). This illustrates the algebraic simplifications often required in calculus to achieve a more elegant form.
Referring to the solution of the problem, the expression \( \frac{2a(x+a)}{(x+a)^2(x-a)} \) was simplified. By expanding \((x+a)(x-a)\) as \(x^2 - a^2\), the expression simplifies to \(\frac{2a}{x^2 - a^2}\). This illustrates the algebraic simplifications often required in calculus to achieve a more elegant form.
- Look out for common factors that can be canceled.
- Algebraic identities like \((a+b)(a-b) = a^2 - b^2\) are invaluable tools.
Verifying Derivatives
Verifying a derivative ensures that the calculations performed were correct. After deriving, you can substitute back and confirm that the resultant derivative matches the known derivative. This check is vital for ensuring that there is no error in reasoning or computation.
In the given problem, after deriving, simplification led to the expression \( \frac{2a}{x^2-a^2} \). This matches the derivative provided in the problem statement. Such verification instills confidence that the derivative was found correctly. Besides, alternative derivative computation methods, such as rules substitution or graphing, can serve as additional verification tools.
In the given problem, after deriving, simplification led to the expression \( \frac{2a}{x^2-a^2} \). This matches the derivative provided in the problem statement. Such verification instills confidence that the derivative was found correctly. Besides, alternative derivative computation methods, such as rules substitution or graphing, can serve as additional verification tools.
- Double-check each solving step to ensure logical consistency.
- Use derivative rules correctly for clear verification.
Other exercises in this chapter
Problem 342
If you are unable to find intersection points analytically in the following exercises, use a calculator. Find the area under \(y=1 / x\) and above the \(x\) -ax
View solution Problem 343
For the following exercises, verify the derivatives and antiderivatives. $$\frac{d}{d x} \ln \left(x+\sqrt{x^{2}+1}\right)=\frac{1}{\sqrt{1+x^{2}}}$$
View solution Problem 345
For the following exercises, verify the derivatives and antiderivatives. $$\frac{d}{d x} \ln \left(\frac{1+\sqrt{1-x^{2}}}{x}\right)=-\frac{1}{x \sqrt{1-x^{2}}}
View solution Problem 346
For the following exercises, verify the derivatives and antiderivatives. $$\frac{d}{d x} \ln \left(x+\sqrt{x^{2}-a^{2}}\right)=\frac{1}{\sqrt{x^{2}-a^{2}}}$$
View solution