Problem 57
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.) $$ g(t)=\frac{10 \log _{4} t}{t} $$
Step-by-Step Solution
Verified Answer
The derivative of \(g(t) = \frac{10 \log _{4} t}{t}\) is \(g'(t) = \frac{10}{t \ln 4} - \frac{10 \log _{4} t}{t^{2}}\).
1Step 1: Identify the Function Type
In the given exercise, the function involved is a quotient of two functions, i.e., the quotient of \(10 \log _{4} t\) and \(t\). Hence, we need to apply the quotient rule of differentiation in this case.
2Step 2: Apply the Quotient Rule
The quotient rule is given by: \(\frac{d}{dx}[\frac{u}{v}] = \frac{vu'-uv'}{v^{2}}\). Now let's identify \(u = 10 \log _{4} t\), \(v = t\) and their derivatives \(u' = \frac{10}{t \ln 4}\) and \(v' = 1\). Apply these into the quotient rule formula, we get: \(g'(t) = \frac{t(\frac{10}{t \ln 4}) - (10 \log _{4} t)(1)}{{t^{2}}}\)
3Step 3: Simplify the Result
Upon simplifying the above equation, we get: \(g'(t) = \frac{10}{\ln 4 \cdot t} - \frac{10 \log _{4} t}{t^{2}} = \frac{10}{t \ln 4} - \frac{10 \log _{4} t}{t^{2}}\). This is the derivative of the function \(g(t) = \frac{10 \log _{4} t}{t}\)
Key Concepts
Quotient Rule DifferentiationLogarithmic DifferentiationSimplifying Derivatives
Quotient Rule Differentiation
Differentiating functions that are formed as a ratio of two other functions—such as \( g(t) = \frac{10 \log_{4} t}{t} \)— requires an understanding of the quotient rule.
The quotient rule states that the derivative of a function \( u/v \) is \( \frac{vu' - uv'}{v^2} \) where \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \) respectively. In the context of our example, \( u \) represents the numerator \( 10 \log_{4} t \) and \( v \) the denominator \( t \) of the given function.
To differentiate this using the quotient rule, we first find the derivatives of \( u \) and \( v \) :
The quotient rule states that the derivative of a function \( u/v \) is \( \frac{vu' - uv'}{v^2} \) where \( u' \) and \( v' \) are the derivatives of \( u \) and \( v \) respectively. In the context of our example, \( u \) represents the numerator \( 10 \log_{4} t \) and \( v \) the denominator \( t \) of the given function.
To differentiate this using the quotient rule, we first find the derivatives of \( u \) and \( v \) :
- The derivative of \( u = 10 \log_{4} t \) is \( u' = \frac{10}{t \ln(4)} \) because the derivative of a logarithm to a certain base is \( 1/(x\ln(\text{base})) \).
- The derivative of \( v = t \) is simply \( v' = 1 \) since the derivative of \( t \) with respect to itself is \( 1 \) .
Logarithmic Differentiation
When faced with differentiating functions that contain logarithms, such as \( 10 \log_{4} t \) in our function \( g(t) \) , logarithmic differentiation is an extremely useful technique.
Logarithmic differentiation relies on the properties of logarithms to simplify the differentiation process. It often involves taking the natural logarithm (ln) of both sides of an equation and then differentiating implicitly. This strategy is particularly effective when the function is a product or quotient whose logarithm can be rewritten as a sum or difference.
For instance, the logarithmic derivative of \( y = \log_{4} x \) can be found by first converting the log to a natural log using the change of base formula, giving us \( y = \frac{\ln x}{\ln 4} \), then differentiating implicitly. One important thing to note is that the derivative of \( \ln x \) is \( 1/x \) , which simplifies the process considerably. Mastering logarithmic differentiation allows students to handle more complicated expressions and is a testament to the power of using algebra to aid in calculus.
Logarithmic differentiation relies on the properties of logarithms to simplify the differentiation process. It often involves taking the natural logarithm (ln) of both sides of an equation and then differentiating implicitly. This strategy is particularly effective when the function is a product or quotient whose logarithm can be rewritten as a sum or difference.
For instance, the logarithmic derivative of \( y = \log_{4} x \) can be found by first converting the log to a natural log using the change of base formula, giving us \( y = \frac{\ln x}{\ln 4} \), then differentiating implicitly. One important thing to note is that the derivative of \( \ln x \) is \( 1/x \) , which simplifies the process considerably. Mastering logarithmic differentiation allows students to handle more complicated expressions and is a testament to the power of using algebra to aid in calculus.
Simplifying Derivatives
Simplifying derivatives is akin to cleaning up the clutter: it involves reducing complex expressions to their most basic form for ease of understanding or further computation.
In computing the derivative of \( g(t) = \frac{10 \log_{4} t}{t} \) , once we apply the quotient rule, we're left with an expression that may still appear complex. The initial derivative contains terms like \( \frac{10}{t\ln 4} \) and \( \frac{10 \log_{4} t}{t^{2}} \). Simplifying involves algebraic manipulation, such as cancelling terms where possible and combining like terms.
For instance, we notice that \( t \) in the numerator and denominator can be simplified, and constants can be factored out. The goal is to get to the simplest form the derivative can take—which, for our function, ends up being: \( \frac{10}{t \ln 4} - \frac{10 \log_{4} t}{t^{2}} \).
Simplification makes the result more interpretable and is a crucial final step in calculus, ensuring that the derivatives are ready for application, whether that's in graphing, optimization problems, or in finding integrals.
In computing the derivative of \( g(t) = \frac{10 \log_{4} t}{t} \) , once we apply the quotient rule, we're left with an expression that may still appear complex. The initial derivative contains terms like \( \frac{10}{t\ln 4} \) and \( \frac{10 \log_{4} t}{t^{2}} \). Simplifying involves algebraic manipulation, such as cancelling terms where possible and combining like terms.
For instance, we notice that \( t \) in the numerator and denominator can be simplified, and constants can be factored out. The goal is to get to the simplest form the derivative can take—which, for our function, ends up being: \( \frac{10}{t \ln 4} - \frac{10 \log_{4} t}{t^{2}} \).
Simplification makes the result more interpretable and is a crucial final step in calculus, ensuring that the derivatives are ready for application, whether that's in graphing, optimization problems, or in finding integrals.
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