Show All Work11. (5 Points) Find Y' Where Y = 5x + 6x + 4
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Understanding the Problem: Finding the Derivative of Y = 5x + 6x + 4
In calculus, one of the fundamental tasks is to find the derivative of a function. Here, we are asked to find the derivative of the function:
\[
Y = 5x + 6x + 4
\]
This problem involves basic rules of differentiation, especially the power rule and the constant rule. To approach this systematically, we need to understand what the derivative represents, how to differentiate each term, and how to combine the results.
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Breaking Down the Function
Step 1: Simplify the function
Before differentiating, it is helpful to simplify the function:
\[
Y = 5x + 6x + 4
\]
Since both terms involving \(x\) are similar, we can combine like terms:
\[
Y = (5x + 6x) + 4 = 11x + 4
\]
This makes differentiation straightforward, as the function is now in the form:
\[
Y = 11x + 4
\]
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Step 2: Understand the components
The simplified function contains:
- A linear term \(11x\),
- A constant term \(4\).
The derivatives of these types of functions follow well-known rules, which we'll explore in detail.
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Differentiation Rules Applied
Rule 1: Power Rule
The power rule states:
\[
\frac{d}{dx} [ax^n] = a \times n x^{n-1}
\]
For linear terms like \(ax\), where \(n=1\), it simplifies to:
\[
\frac{d}{dx} [ax] = a
\]
Rule 2: Constant Rule
The derivative of a constant term is zero:
\[
\frac{d}{dx} [c] = 0
\]
Applying these rules to our simplified function:
\[
Y = 11x + 4
\]
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Calculating the Derivative of Y
Step 1: Differentiate the term \(11x\)
Using the constant multiple rule and the fact that the derivative of \(x\) is 1:
\[
\frac{d}{dx} [11x] = 11 \times \frac{d}{dx} [x] = 11 \times 1 = 11
\]
Step 2: Differentiate the constant \(4\)
\[
\frac{d}{dx} [4] = 0
\]
Step 3: Combine the derivatives
Adding the derivatives of each term:
\[
Y' = 11 + 0 = 11
\]
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Final Answer: The Derivative of Y
\[
\boxed{
Y' = 11
}
\]
This indicates that the slope of the function \(Y = 11x + 4\) at any point is 11, meaning it is a straight line with a constant rate of change.
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Additional Insights into Derivatives and Their Significance
What Does the Derivative Represent?
In calculus, the derivative of a function at a specific point measures the rate at which the function's value changes with respect to its input. For a linear function such as \(Y = 11x + 4\), the derivative is constant, reflecting a uniform rate of change.
Geometric Interpretation
The derivative \(Y' = 11\) represents the slope of the line described by the function \(Y\). Since the line is straight and has a constant slope, the derivative remains the same across the entire domain.
Practical Applications
Understanding the derivative in such functions is essential in various fields:
- Physics: Calculating velocity as the rate of change of position.
- Economics: Finding marginal cost or revenue.
- Engineering: Analyzing rates of change in systems.
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Common Mistakes to Avoid When Differentiating Similar Functions
- Forgetting to simplify the function first: Always combine like terms to make differentiation easier.
- Misapplying the power rule: Remember that the power rule applies to terms with variables raised to a power.
- Neglecting to differentiate constants: The derivative of a constant is zero.
- Ignoring the constant multiple rule: When multiplying a constant by a variable, differentiate the variable term and multiply the result by the constant.
Summary and Key Takeaways
- The original function \(Y = 5x + 6x + 4\) simplifies to \(Y = 11x + 4\).
- The derivative of a linear function \(ax + c\) is simply \(a\).
- Therefore, the derivative of \(Y\) is \(Y' = 11\).
- The process involves applying basic differentiation rules: the power rule and the constant rule.
- Understanding derivatives helps analyze the behavior of functions and their rates of change.
Additional Practice Problems
To strengthen understanding of derivatives in similar contexts, consider practicing:
- Find the derivative of \(Y = 3x^2 + 7x + 5\).
- Differentiate \(Y = 4x^3 - 2x + 1\).
- Compute the derivative of \(Y = 9x - 4\).
- Find the derivative of \(Y = \frac{1}{2}x + 3\).
Practicing these will solidify your understanding of differentiation rules and their applications.
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Conclusion
In conclusion, the process of finding the derivative of a function like \(Y = 5x + 6x + 4\) involves simplifying the expression and applying fundamental derivative rules. The derivative, in this case, is a constant 11, indicating a constant rate of change. Mastering these differentiation techniques is crucial for solving more complex calculus problems and understanding how functions behave across different domains.
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References for Further Reading
- Calculus Textbooks: "Calculus: Early Transcendentals" by James Stewart.
- Online Resources: Khan Academy's Calculus Course.
- Mathematics Websites: Paul's Online Math Notes on Derivatives.
- Educational Platforms: Brilliant.org's Differential Calculus Modules.
