Find Derivatives Of F(x) = 3 + 5/x + 7/x^2 A Step-by-Step Guide

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This article delves into the step-by-step solution of finding the first and second derivatives of the function ${ f(x) = 3 + \frac{5}{x} + \frac{7}{x^2} }$, and then evaluating these derivatives at ${ x = 4 }$. We will explore the fundamental concepts of calculus involved, providing a comprehensive guide suitable for students and enthusiasts alike. Understanding derivatives is crucial in various fields, including physics, engineering, and economics, as they represent rates of change. This exploration not only solves the given problem but also reinforces the core principles of differentiation.

Finding the First Derivative $f^{\prime}(x)$

To find the first derivative, ${ f^{\prime}(x) }$, we need to differentiate the given function ${ f(x) = 3 + \frac{5}{x} + \frac{7}{x^2} }$ with respect to ${ x }$. This involves applying the power rule and the constant multiple rule of differentiation. Let’s break down the process step by step.

First, it's beneficial to rewrite the function using negative exponents to make differentiation easier. The function becomes:

${ f(x) = 3 + 5x^{-1} + 7x^{-2} }$

Now we can differentiate each term separately. The derivative of a constant is zero, and we apply the power rule ${ \frac{d}{dx}(x^n) = nx^{n-1} }$ to the other terms. The power rule is a cornerstone of differential calculus, allowing us to find the rate of change of power functions. It's essential to grasp this rule thoroughly, as it appears in countless calculus problems. The constant multiple rule, which states that the derivative of a constant times a function is the constant times the derivative of the function, also plays a crucial role here.

The derivative of the constant term 3 is 0. For the term ${ 5x^{-1} }$, we apply the power rule and the constant multiple rule. We multiply the coefficient 5 by the exponent -1 and reduce the exponent by 1: ${ \frac{d}{dx}(5x^{-1}) = 5(-1)x^{-1-1} = -5x^{-2} }$

Similarly, for the term ${ 7x^{-2} }$, we apply the same rules: ${ \frac{d}{dx}(7x^{-2}) = 7(-2)x^{-2-1} = -14x^{-3} }$

Combining these results, we find the first derivative:

${ f^{\prime}(x) = 0 - 5x^{-2} - 14x^{-3} }$

This can be rewritten in a more conventional form using fractions:

${ f^{\prime}(x) = -\frac{5}{x^2} - \frac{14}{x^3} }$

This expression represents the instantaneous rate of change of the function ${ f(x) }$ at any point ${ x }$. Understanding how to derive this expression is fundamental to calculus and its applications.

Key Concepts Applied

• Power Rule: ${ \frac{d}{dx}(x^n) = nx^{n-1} }$
• Constant Multiple Rule: ${ \frac{d}{dx}[cf(x)] = c \frac{d}{dx}f(x) }$, where ${ c }$ is a constant.
• Derivative of a Constant: The derivative of a constant is 0.

Evaluating $f^{\prime}(4)$

Now that we have found the first derivative, ${ f^{\prime}(x) = -\frac{5}{x^2} - \frac{14}{x^3} }$, we can evaluate it at ${ x = 4 }$ to find the instantaneous rate of change of the function at this specific point. Evaluating derivatives at specific points is a common practice in calculus, often used to find slopes of tangent lines, velocities at certain times, or marginal costs in economics.

To find ${ f^{\prime}(4) }$, we simply substitute ${ x = 4 }$ into the expression for ${ f^{\prime}(x) }$:

${ f^{\prime}(4) = -\frac{5}{(4)^2} - \frac{14}{(4)^3} }$

Now we perform the calculations. First, we calculate the powers of 4:

${ (4)^2 = 16 }$

${ (4)^3 = 64 }$

Substituting these values back into the expression, we get:

${ f^{\prime}(4) = -\frac{5}{16} - \frac{14}{64} }$

To add these fractions, we need a common denominator. The least common multiple of 16 and 64 is 64. So, we convert the first fraction to have a denominator of 64:

${ -\frac{5}{16} = -\frac{5 \times 4}{16 \times 4} = -\frac{20}{64} }$

Now we can add the fractions:

${ f^{\prime}(4) = -\frac{20}{64} - \frac{14}{64} = -\frac{20 + 14}{64} = -\frac{34}{64} }$

Finally, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

${ f^{\prime}(4) = -\frac{34 \div 2}{64 \div 2} = -\frac{17}{32} }$

Thus, the value of the first derivative at ${ x = 4 }$ is ${ -\frac{17}{32} }$. This value represents the slope of the tangent line to the curve of ${ f(x) }$ at the point where ${ x = 4 }$. Understanding how to evaluate derivatives at specific points is crucial for solving optimization problems and analyzing the behavior of functions.

Steps for Evaluation

1. Substitute the given value of ${ x }$ into the derivative expression.
2. Calculate the powers and perform the arithmetic operations.
3. Simplify the resulting fraction, if possible.

Finding the Second Derivative $f^{\prime \prime}(x)$

To find the second derivative, ${ f^{\prime \prime}(x) }$, we need to differentiate the first derivative, ${ f^{\prime}(x) }$, with respect to ${ x }$. We found earlier that ${ f^{\prime}(x) = -\frac{5}{x^2} - \frac{14}{x^3} }$. Again, rewriting with negative exponents will simplify the differentiation process. The second derivative provides information about the concavity of the original function. A positive second derivative indicates that the function is concave up, while a negative second derivative indicates that the function is concave down. Understanding concavity is essential for sketching curves and solving optimization problems.

Rewriting the first derivative with negative exponents, we have:

${ f^{\prime}(x) = -5x^{-2} - 14x^{-3} }$

Now we differentiate each term separately using the power rule and the constant multiple rule. For the first term, ${ -5x^{-2} }$, we apply the power rule:

${ \frac{d}{dx}(-5x^{-2}) = -5(-2)x^{-2-1} = 10x^{-3} }$

For the second term, ${ -14x^{-3} }$, we also apply the power rule:

${ \frac{d}{dx}(-14x^{-3}) = -14(-3)x^{-3-1} = 42x^{-4} }$

Combining these results, we find the second derivative:

${ f^{\prime \prime}(x) = 10x^{-3} + 42x^{-4} }$

This can be rewritten in a more conventional form using fractions:

${ f^{\prime \prime}(x) = \frac{10}{x^3} + \frac{42}{x^4} }$

This expression represents the rate of change of the first derivative, which provides insights into the curvature of the original function. A positive second derivative indicates that the rate of change of the slope is increasing, while a negative second derivative indicates that the rate of change of the slope is decreasing. This information is crucial for determining maximum and minimum points of a function.

Derivative Rules Revisited

• Power Rule: ${ \frac{d}{dx}(x^n) = nx^{n-1} }$
• Constant Multiple Rule: ${ \frac{d}{dx}[cf(x)] = c \frac{d}{dx}f(x) }$, where ${ c }$ is a constant.

Evaluating $f^{\prime \prime}(4)$

Now that we have found the second derivative, ${ f^{\prime \prime}(x) = \frac{10}{x^3} + \frac{42}{x^4} }$, we can evaluate it at ${ x = 4 }$ to understand the concavity of the function at this point. Evaluating the second derivative at a specific point can help us determine whether the function is concave up or concave down at that point, which is useful for optimization problems and curve sketching.

To find ${ f^{\prime \prime}(4) }$, we substitute ${ x = 4 }$ into the expression for ${ f^{\prime \prime}(x) }$:

${ f^{\prime \prime}(4) = \frac{10}{(4)^3} + \frac{42}{(4)^4} }$

First, we calculate the powers of 4:

${ (4)^3 = 64 }$

${ (4)^4 = 256 }$

Substituting these values back into the expression, we get:

${ f^{\prime \prime}(4) = \frac{10}{64} + \frac{42}{256} }$

To add these fractions, we need a common denominator. The least common multiple of 64 and 256 is 256. So, we convert the first fraction to have a denominator of 256:

${ \frac{10}{64} = \frac{10 \times 4}{64 \times 4} = \frac{40}{256} }$

Now we can add the fractions:

${ f^{\prime \prime}(4) = \frac{40}{256} + \frac{42}{256} = \frac{40 + 42}{256} = \frac{82}{256} }$

Finally, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

${ f^{\prime \prime}(4) = \frac{82 \div 2}{256 \div 2} = \frac{41}{128} }$

Thus, the value of the second derivative at ${ x = 4 }$ is ${ \frac{41}{128} }$. Since this value is positive, the function is concave up at ${ x = 4 }$. This means that the rate of change of the slope is increasing at this point, indicating a curve that opens upwards.

Significance of the Second Derivative

• Concavity: If ${ f^{\prime \prime}(x) > 0 }$, the function is concave up.
• Inflection Points: Points where ${ f^{\prime \prime}(x) }$ changes sign are inflection points.
• Optimization: Second derivative test can be used to determine local maxima and minima.

Conclusion

In summary, we have successfully found the first and second derivatives of the function ${ f(x) = 3 + \frac{5}{x} + \frac{7}{x^2} }$ and evaluated them at ${ x = 4 }$. We found that:

• The first derivative is ${ f^{\prime}(x) = -\frac{5}{x^2} - \frac{14}{x^3} }$.
• The value of the first derivative at ${ x = 4 }$ is ${ f^{\prime}(4) = -\frac{17}{32} }$.
• The second derivative is ${ f^{\prime \prime}(x) = \frac{10}{x^3} + \frac{42}{x^4} }$.
• The value of the second derivative at ${ x = 4 }$ is ${ f^{\prime \prime}(4) = \frac{41}{128} }$.

This exercise illustrates the fundamental principles of differentiation and their application in evaluating functions. Understanding derivatives is crucial for various applications in mathematics, physics, engineering, and other fields. By mastering these concepts, one can gain valuable insights into the behavior of functions and their rates of change. The process of finding derivatives involves applying rules such as the power rule, constant multiple rule, and sum/difference rule. Evaluating derivatives at specific points provides information about the function's behavior at those points, such as its slope and concavity. This comprehensive guide should serve as a valuable resource for anyone looking to deepen their understanding of calculus.