Problem 52
Question
Finding a Derivative In Exercises \(37-58\) , find the derivative of the function. (Hint: In some exercises, you may find it helpful to apply logarithmic properties before differentiating.) $$ f(x)=\log _{2} \sqrt[3]{2 x+1} $$
Step-by-Step Solution
Verified Answer
The derivative of the function \(f(x) = \log _{2} \sqrt[3]{2 x+1}\) is \(f'(x) = \frac{2}{3ln(2)(2x+1)}\)
1Step 1: Simplify the Function
Using the property of logarithms \(log_b(m^n) = n*log_b(m)\), the function simplifies to: \(f(x) = \frac{1}{3} \log_{2}(2x+1)\)
2Step 2: Convert the Base of the Logarithm
The base 2 logarithm can be difficult to deal with so we switch it to a natural log, which is easier to differentiate. Also using the formula \(log_b(a) = \frac{ln(a)}{ln(b)}\), we have: \(f(x) = \frac{1}{3} \frac{ln(2x+1)}{ln(2)}\) which simplifies to \(f(x) = \frac{ln(2x+1)}{3ln(2)}\)
3Step 3: Find the Derivative
Now we apply the chain rule, which states that the derivative of a function composed of another function is the derivative of the outer function times the derivative of the inner function. Begin with differentiating the outer function and then the inner function. Applying the chain rule, the derivative \(f'(x)\) becomes \(f'(x) = \frac{1}{3ln(2)} \frac{1}{2x+1} * 2 = \frac{2}{3ln(2)(2x+1)}\)
Key Concepts
Logarithmic DifferentiationChain RuleLogarithm PropertiesNatural Logarithm
Logarithmic Differentiation
Logarithmic differentiation is a useful technique when dealing with complex functions, especially those involving products, quotients, or exponents. It involves taking the natural logarithm of both sides of an equation and using properties of logarithms to simplify the differentiation process. This method is particularly beneficial because it can transform multiplicative or exponential expressions into simpler, additive forms. By applying logarithms, complex functions become easier to handle and differentiate.
In our exercise, taking the logarithm allowed us to express \(f(x) = \log_{2} \sqrt[3]{2x+1}\) in a more tractable form. We transformed it into \(f(x) = \frac{1}{3} \log_{2}(2x+1)\), making the differentiation process much more straightforward.
In our exercise, taking the logarithm allowed us to express \(f(x) = \log_{2} \sqrt[3]{2x+1}\) in a more tractable form. We transformed it into \(f(x) = \frac{1}{3} \log_{2}(2x+1)\), making the differentiation process much more straightforward.
Chain Rule
The chain rule is an essential tool in calculus for finding the derivative of a composite function. It states that the derivative of a function \(g(f(x))\) is the derivative of the outer function \(g\) evaluated at the inner function \(f(x)\), multiplied by the derivative of the inner function \(f(x)\).
In the original problem, the function \(f(x) = \log_{2}\sqrt[3]{2x+1}\) needed differentiation. Upon simplifying, we obtained a composite function, where the derivatives of both the logarithmic and polynomial expressions had to be found. Using the chain rule, we computed the derivative of the natural logarithm of the inner expression \(2x+1\), then multiplied it by the constant from the logarithm conversion. This led us to the final derivative, \(f'(x) = \frac{2}{3\ln(2)(2x+1)}\).
In the original problem, the function \(f(x) = \log_{2}\sqrt[3]{2x+1}\) needed differentiation. Upon simplifying, we obtained a composite function, where the derivatives of both the logarithmic and polynomial expressions had to be found. Using the chain rule, we computed the derivative of the natural logarithm of the inner expression \(2x+1\), then multiplied it by the constant from the logarithm conversion. This led us to the final derivative, \(f'(x) = \frac{2}{3\ln(2)(2x+1)}\).
Logarithm Properties
Logarithm properties are mathematical rules that help simplify the process of working with logarithms. These properties make it easier to solve or differentiate equations involving logarithms. Key properties include:
This simplification was pivotal for subsequent steps, making the application of differentiation techniques like the chain rule much more efficient and straightforward.
- Product Property: \(\log_b(mn) = \log_b(m) + \log_b(n)\)
- Quotient Property: \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)
- Power Property: \(\log_b(m^n) = n \cdot \log_b(m)\)
This simplification was pivotal for subsequent steps, making the application of differentiation techniques like the chain rule much more efficient and straightforward.
Natural Logarithm
The natural logarithm, denoted as \(\ln\), is a logarithm to the base \(e\), where \(e \approx 2.71828\). It is a fundamental function in calculus due to its unique derivative properties. When differentiating natural logarithms, the derivative of \(\ln(x)\) is simply \(\frac{1}{x}\).
In this exercise, converting the logarithm from base 2 to a natural logarithm simplified the differentiation significantly. Using the change of base formula, \(\log_b(a) = \frac{\ln(a)}{\ln(b)}\), we translated the function into a natural logarithm form. This allowed us to utilize the simpler derivative formula of the natural logarithm, which is crucial in solving for \(f'(x)\).
Thus, with natural logarithm differentiation, dealing with complex bases like 2 becomes much more manageable in calculus.
In this exercise, converting the logarithm from base 2 to a natural logarithm simplified the differentiation significantly. Using the change of base formula, \(\log_b(a) = \frac{\ln(a)}{\ln(b)}\), we translated the function into a natural logarithm form. This allowed us to utilize the simpler derivative formula of the natural logarithm, which is crucial in solving for \(f'(x)\).
Thus, with natural logarithm differentiation, dealing with complex bases like 2 becomes much more manageable in calculus.
Other exercises in this chapter
Problem 52
Find the derivative of the function. \(y=\frac{1}{2}\left[x \sqrt{4-x^{2}}+4 \arcsin \left(\frac{x}{2}\right)\right]\)
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In Exercises 43–54, find the indefinite integral. $$ \int \operatorname{sech}^{3} x \tanh x d x $$
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Finding a Derivative In Exercises \(33-54,\) find the derivative. $$ y=e^{2 x} \tan 2 x $$
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In Exercises 41–64, find the derivative of the function. $$ f(x)=\ln \left(\frac{2 x}{x+3}\right) $$
View solution