Problem 12
Question
Find the value of the derivative of the function at the given point. State which differentiation rule you used to find the derivative. $$ f(x)=\frac{3 x}{x^{2}+4} \quad\left(-1,-\frac{3}{5}\right) $$
Step-by-Step Solution
Verified Answer
The derivative of the function \( f(x) = \frac{3x}{x^2 + 4} \) at the point \(-1,-\frac{3}{5}\) is 1. The Quotient Rule was used to determine this.
1Step 1: Identify the Quotient Rule
The Quotient Rule for differentiation states that the derivative of the quotient of two functions is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator. For a function in the form \( f(x)=\frac{g(x)}{h(x)} \), the derivative \( f'(x)=\frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} \). In this case, \( g(x) = 3x \) and \( h(x) = x^2 + 4 \).
2Step 2: Apply the Quotient Rule
Calculate the derivatives \( g'(x) \) and \( h'(x) \). For \( g(x) = 3x \), its derivative \( g'(x) = 3 \). For \( h(x) = x^2 + 4 \), its derivative \( h'(x) = 2x \). Substituting these values into the quotient rule equation gives: \( f'(x) = \frac{3(x^2+4) - 3x(2x)}{(x^2 + 4)^2} = \frac{3x^2+12 - 6x^2}{(x^2 + 4)^2} = \frac{-3x^2+12}{(x^2 + 4)^2} \)
3Step 3: Calculate the Value of the Derivative at the Given Point
By substituting \( x = -1 \) into \( f'(x) \), you will get: \( f'(-1) = \frac{-3(-1)^2+12}{(-1^2 + 4)^2} = \frac{-3+12}{(-1 + 4)^2} = \frac{9}{9} = 1 \).
Key Concepts
DifferentiationDerivativeCalculus
Differentiation
Differentiation is a fundamental concept in calculus that involves finding the rate at which a function changes at any given point. It allows us to compute the slope or the gradient of a function's curve at any point, which is particularly useful in understanding the behavior of functions.
Differentiation applies various rules to compute derivatives, which are the results of these calculations. One of these rules is the Quotient Rule, used when differentiating a function that is the quotient (or division) of two other functions. In our original exercise, we applied the quotient rule to differentiate the function \( f(x) = \frac{3x}{x^2 + 4} \).
With the Quotient Rule, the differentiation process becomes straightforward, offering a simple mechanism to tackle division-based functions. Differentiation is largely used for practical applications including:
Differentiation applies various rules to compute derivatives, which are the results of these calculations. One of these rules is the Quotient Rule, used when differentiating a function that is the quotient (or division) of two other functions. In our original exercise, we applied the quotient rule to differentiate the function \( f(x) = \frac{3x}{x^2 + 4} \).
With the Quotient Rule, the differentiation process becomes straightforward, offering a simple mechanism to tackle division-based functions. Differentiation is largely used for practical applications including:
- Calculating speed, velocity, and acceleration in physics.
- Finding the slope of a curve in geometry.
- Maximizing or minimizing functions in economics and business.
Derivative
The derivative is essentially the result of the differentiation process. It measures how a function's output changes as we change its input. More formally, if \( y = f(x) \) is a function, then the derivative, \( f'(x) \), represents the instantaneous rate of change of \( y \) with respect to \( x \).
This makes derivatives powerful tools in analyzing real-world phenomena that involve change, such as the change in cost relative to production levels, or the change in position relative to time. In the context of our original exercise, we found the derivative of \( f(x) = \frac{3x}{x^2 + 4} \) using the Quotient Rule. By doing so, we were able to calculate the derivative at a specific point, \((-1, -\frac{3}{5})\), which illustrates how derivatives provide detailed insights into the behavior of functions at particular points.
Key applications of derivatives include:
This makes derivatives powerful tools in analyzing real-world phenomena that involve change, such as the change in cost relative to production levels, or the change in position relative to time. In the context of our original exercise, we found the derivative of \( f(x) = \frac{3x}{x^2 + 4} \) using the Quotient Rule. By doing so, we were able to calculate the derivative at a specific point, \((-1, -\frac{3}{5})\), which illustrates how derivatives provide detailed insights into the behavior of functions at particular points.
Key applications of derivatives include:
- Analyzing rates of change.
- Determining local maxima and minima in graphs.
- Simplifying complex dynamical systems to solve differential equations.
Calculus
Calculus is the mathematical study of continuous change and is divided into two main branches: differential calculus, which cares predominantly about rates of change (differentiation), and integral calculus, which focuses on accumulation of quantities.
Central to calculus is the concept of limits, which allows us to define derivatives and integrals in rigorous terms. The power of calculus lies in its ability to model and solve real-world problems across various fields, from engineering to economics.
In our particular problem, we used principles from differential calculus to find derivatives using the Quotient Rule. Calculus tools like this enable us to compute exact changes and predict behavior in virtually any measurable system, from population growth models to electrical circuits. With a good understanding of calculus, one gains the ability to relate various quantities and interpret data trends accurately.
How calculus is used:
Central to calculus is the concept of limits, which allows us to define derivatives and integrals in rigorous terms. The power of calculus lies in its ability to model and solve real-world problems across various fields, from engineering to economics.
In our particular problem, we used principles from differential calculus to find derivatives using the Quotient Rule. Calculus tools like this enable us to compute exact changes and predict behavior in virtually any measurable system, from population growth models to electrical circuits. With a good understanding of calculus, one gains the ability to relate various quantities and interpret data trends accurately.
How calculus is used:
- To calculate maximum efficiency points in processes.
- To find the best trajectory paths in aerospace engineering.
- To determine economic cost functions and profit optimization.
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