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Magazine Chapter 27: Problem 5
Find the derivatives of the given functions. $$y=4 \log _{5}(3-x)$$
Short Answer
Expert verified
The derivative is \(-\frac{4}{(3-x) \ln(5)}\)."
Step by step solution
01
Recall the Derivative of a Logarithmic Function
The derivative of a natural logarithmic function \( y = \ \ln(u) \) is given by \( \frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx} \). When dealing with a logarithm of a different base, the formula changes slightly. For a logarithm with base \( a \), such as \( f(x) = \log_a(u) \), the derivative is: \( f'(x) = \frac{1}{u \ln(a)} \cdot \frac{du}{dx} \).
02
Identify the Components of the Function
In the given function, \( y = 4 \log_{5} (3-x) \), identify \( u \) as \( 3-x \) and the base \( a \) as \( 5 \). We also notice the constant multiplier of \( 4 \) in front of the logarithm.
03
Apply the Constant Multiple Rule
The derivative of a constant times a function is the constant times the derivative of the function. So, we have \( \frac{dy}{dx} = 4 \cdot \frac{d}{dx} \log_{5} (3-x) \).
04
Differentiate the Logarithmic Function
Now, using the formula for the derivative of a logarithmic function with base \( a \): \[\frac{d}{dx} \log_{5} (3-x) = \frac{1}{(3-x) \ln(5)} \cdot \frac{d}{dx}(3-x)\]Calculate \( \frac{d}{dx}(3-x) \), which is \(-1\).
05
Combine the Derivatives
Combine the results from Steps 3 and 4: \[\frac{dy}{dx} = 4 \cdot \left( \frac{-1}{(3-x) \ln(5)} \right) = -\frac{4}{(3-x) \ln(5)}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithmic Functions
Logarithmic functions play a pivotal role in calculus, especially when dealing with growth and decay processes. A basic logarithmic function has the form \( y = \log_a(x) \), where \( a \) is the base, and \( x \) is the argument of the logarithm.
Logarithms are essentially the inverse of exponentiation. For example, if we say \( \log_a(b) = c \), it means that \( a^c = b \). This relationship allows us to convert between exponential and logarithmic forms, giving us flexibility in problem-solving.
When differentiating logarithmic functions, it becomes important to recognize its base. If the base is not the natural number \( e \), the derivative formula must incorporate the conversion factor \( \ln(a) \), where \( a \) is the base. This ensures the derivative considers the base's effect, making it widely applicable across different scenarios involving logarithms.
Logarithms are essentially the inverse of exponentiation. For example, if we say \( \log_a(b) = c \), it means that \( a^c = b \). This relationship allows us to convert between exponential and logarithmic forms, giving us flexibility in problem-solving.
When differentiating logarithmic functions, it becomes important to recognize its base. If the base is not the natural number \( e \), the derivative formula must incorporate the conversion factor \( \ln(a) \), where \( a \) is the base. This ensures the derivative considers the base's effect, making it widely applicable across different scenarios involving logarithms.
Constant Multiple Rule
The constant multiple rule is a handy tool in differentiation. It states that the derivative of a constant multiplied by a function is just the constant multiplied by the derivative of that function.
For instance, if you have a function \( y = c \, f(x) \), where \( c \) is a constant, then the derivative \( \frac{dy}{dx} = c \cdot \frac{df(x)}{dx} \). It's like saying the constant tagging along doesn't change how we handle the differentiation process; it just scales the result.
This rule simplifies differentiating functions with constant coefficients, making calculations straightforward and efficient. Whenever you see a constant in a differentiation problem, remember it simply carries through as a multiplier, streamlining your calculations.
For instance, if you have a function \( y = c \, f(x) \), where \( c \) is a constant, then the derivative \( \frac{dy}{dx} = c \cdot \frac{df(x)}{dx} \). It's like saying the constant tagging along doesn't change how we handle the differentiation process; it just scales the result.
This rule simplifies differentiating functions with constant coefficients, making calculations straightforward and efficient. Whenever you see a constant in a differentiation problem, remember it simply carries through as a multiplier, streamlining your calculations.
Change of Base in Logarithms
Sometimes you'll deal with logarithms that aren't in the base you prefer or need, like the natural logarithm base \( e \). The change of base formula allows us to transform any logarithm into a more workable form.
This formula is: \( \log_b(x) = \frac{\ln(x)}{\ln(b)} \). This means you can express the logarithm with any base in terms of natural logarithms, which are often easier to handle, especially when differentiating.
The essence of this change-of-base method is to facilitate operations. When working with derivatives, using natural logs simplifies many aspects of the process because the derivative of \( \ln(x) \) is a well-known and manageable form. This adaptability makes change of base a crucial concept in both theoretical and practical applications.
This formula is: \( \log_b(x) = \frac{\ln(x)}{\ln(b)} \). This means you can express the logarithm with any base in terms of natural logarithms, which are often easier to handle, especially when differentiating.
The essence of this change-of-base method is to facilitate operations. When working with derivatives, using natural logs simplifies many aspects of the process because the derivative of \( \ln(x) \) is a well-known and manageable form. This adaptability makes change of base a crucial concept in both theoretical and practical applications.
