Problem 29
Question
Find the derivative. Assume \(a, b, c, k\) are constants. $$f(x)=k x^{2}$$
Step-by-Step Solution
Verified Answer
The derivative of \(f(x) = kx^2\) is \(2kx\).
1Step 1: Identify the power rule
The function given is in the form of a power rule function: \[ f(x) = kx^n \]where \(n\) is an integer. In this case, \(n = 2\).
2Step 2: Apply the power rule
The power rule for differentiation states that if \(f(x) = ax^n\), then the derivative \(f'(x)\) is: \[ f'(x) = anx^{n-1} \].For \(f(x) = kx^2\), the derivative is computed as:\[ f'(x) = k imes 2 imes x^{2-1} = 2kx \].
3Step 3: Write the derivative
After applying the power rule, the derivative of the function \(f(x) = kx^2\) is:\[ f'(x) = 2kx \].
Key Concepts
Power RuleDifferentiationCalculus
Power Rule
The power rule is a fundamental tool in calculus for finding derivatives quickly and easily. When you have a function in the form of \( f(x) = ax^n \), the power rule allows you to find its derivative by focusing on the exponent \( n \) of \( x \).
The main idea here is simple: take that exponent and multiply it by the coefficient \( a \), then reduce the exponent by one. This results in the derivative: \( f'(x) = anx^{n-1} \).
By doing this, you efficiently determine the rate at which the function changes at any given point. In the given exercise, with \( f(x) = kx^2 \), we apply the power rule to get \( f'(x) = 2kx \). This means, for every unit change in \( x \), the function changes by \( 2kx \) units.
The main idea here is simple: take that exponent and multiply it by the coefficient \( a \), then reduce the exponent by one. This results in the derivative: \( f'(x) = anx^{n-1} \).
By doing this, you efficiently determine the rate at which the function changes at any given point. In the given exercise, with \( f(x) = kx^2 \), we apply the power rule to get \( f'(x) = 2kx \). This means, for every unit change in \( x \), the function changes by \( 2kx \) units.
Differentiation
Differentiation is the process of finding a derivative, which provides us with a powerful tool to understand and analyze how a function behaves. It tells us how a function changes or reacts as its input, \( x \), changes.
Essentially, the derivative represents the slope of the tangent line to the graph of the function at any given point on the curve. Differentiation allows us to identify critical points, where the function reaches minimum or maximum values, and understand the nature of these points better.
In the exercise on \( f(x) = kx^2 \), we use differentiation to get \( f'(x) = 2kx \), which reflects how the function changes its direction and behavior. This knowledge is crucial in various fields, from physics to economics, whenever it is necessary to predict changes and outcomes.
Essentially, the derivative represents the slope of the tangent line to the graph of the function at any given point on the curve. Differentiation allows us to identify critical points, where the function reaches minimum or maximum values, and understand the nature of these points better.
In the exercise on \( f(x) = kx^2 \), we use differentiation to get \( f'(x) = 2kx \), which reflects how the function changes its direction and behavior. This knowledge is crucial in various fields, from physics to economics, whenever it is necessary to predict changes and outcomes.
Calculus
Calculus is the branch of mathematics that deals with the study of continuous change. It's divided into two main branches: differentiation, which we discussed, and integration. Together, these concepts form the backbone of calculus, allowing us to analyze complex systems and changes in various scientific fields and everyday life.
Calculus provides the tools for modeling and solving problems involving motion, growth, decay, and other dynamic processes. In this context, finding the derivative of a function like \( f(x) = kx^2 \) is just one aspect of calculus, but it illustrates the powerful role this discipline has in understanding how variables interact.
By mastering calculus, you learn not only to solve mathematical problems but also to apply these solutions to real-world scenarios, from engineering designs to predicting market trends.
Calculus provides the tools for modeling and solving problems involving motion, growth, decay, and other dynamic processes. In this context, finding the derivative of a function like \( f(x) = kx^2 \) is just one aspect of calculus, but it illustrates the powerful role this discipline has in understanding how variables interact.
By mastering calculus, you learn not only to solve mathematical problems but also to apply these solutions to real-world scenarios, from engineering designs to predicting market trends.
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Problem 29
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