Problem 90 Assume that \(f(x)\) and \(g(x)\... [FREE SOLUTION]

Chapter 4: Problem 90

Assume that \(f(x)\) and \(g(x)\) are differentiable at x. Find an expression for the derivative of y in terms of \(f(x), g(x), f^{\prime}(x)\), and \(g^{\prime}(x) .\) \(y=\frac{x^{2}}{f(x)+g(x)}\)

Short Answer

Expert verified

The derivative is \(y' = \frac{2x(f(x) + g(x)) - x^2(f'(x) + g'(x))}{(f(x) + g(x))^2}\)."

Step by step solution

Apply the Quotient Rule

Since the function \(y = \frac{x^2}{f(x) + g(x)}\) is a quotient of two functions, we use the quotient rule for derivatives. The quotient rule states that for a function \(y = \frac{u(x)}{v(x)}\), the derivative \(y'\) is given by \( y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2} \). Identify \(u(x) = x^2\) and \(v(x) = f(x) + g(x)\).

Differentiate the Numerator

Differentiate the numerator function, \(u(x) = x^2\). The derivative of \(x^2\) with respect to \(x\) is \(u'(x) = 2x\).

Differentiate the Denominator

Differentiate the denominator function, \(v(x) = f(x) + g(x)\). Using the sum rule, the derivative is \(v'(x) = f'(x) + g'(x)\).

Substitute into the Quotient Rule

Substitute \(u(x)\), \(u'(x)\), \(v(x)\), and \(v'(x)\) into the quotient rule formula. This gives:\[y' = \frac{(2x)\cdot(f(x) + g(x)) - (x^2)\cdot(f'(x) + g'(x))}{(f(x) + g(x))^2}\]

Simplify the Expression

The expression is already simplified enough, but make sure it is clear:\[y' = \frac{2x(f(x) + g(x)) - x^2(f'(x) + g'(x))}{(f(x) + g(x))^2}\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differentiation

Differentiation is a key concept in calculus that enables us to determine the rate at which a function changes. It is the process that calculates the derivative, which is effectively a measure of how a function's value shifts as its input varies. In simple terms, differentiation answers the question, "How fast is something changing?"

Understanding the

Basics of Differentiation: Differentiation involves applying specific rules to find the derivative of a function.
Function's Behavior: It helps us understand how functions behave, identifying whether they increase, decrease, or remain constant at particular points.
Applications: Differentiation plays a crucial role in fields such as physics, engineering, and economics, wherever optimization and rate of change are important.

When tackling differentiation problems such as the exercise at hand, recognizing the function types and applying appropriate rules like the product rule, chain rule, or quotient rule is essential.

Quotient Rule

The quotient rule is a helpful tool when you're dealing with a function that involves the division of two other functions鈥攕pecifically, the "quotient" of two functions.

To apply the quotient rule, remember the following general formula for a function indicated as \(y = \frac{u(x)}{v(x)}\):

The formula for the derivative, given these functions, is: \(y' = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\).
Key Components: \(u(x)\) is the "top" function, \(v(x)\) is the "bottom". Differentiating each part before substituting into the formula is crucial.
Application: In our exercise, \(u(x) = x^2\) and \(v(x) = f(x) + g(x)\), which makes our formula specific to the exercise's context.

Remember:

Differentiate each part separately.
Complete these derivatives before substitution.
Simplify the expression as the final step.

Using the quotient rule correctly allows us to discern complex functions that involve division with accuracy and clarity.

Derivative

The derivative is a central concept in calculus, representing the instantaneous rate of change of a function with respect to one of its variables. In other words, it's how we formalize the concept of "rate of change" we often discuss in real-life scenarios.

Here's why derivatives are so important:

Function Analysis: The derivative gives us a mathematical tool for analyzing changes in functions and understanding their behavior.
Geometric Interpretation: The derivative at a specific point gives us the slope of the tangent line to the function at that point.
Broad Applications: Derivatives are used across various disciplines to model behavior, optimize processes, and analyze patterns.

In our exercise, finding the derivative involved:

Using differentiation rules to find the rate of change of the function \(y = \frac{x^2}{f(x) + g(x)}\).
Utilizing the quotient rule to manage the complexity of division within the function.

Understanding how to determine derivatives allows for deeper insights into mathematical behavior, paving the way for their broad utilization in practical applications.

91影视

Assume that \(f(x)\) and \(g(x)\) are differentiable at x. Find an expression for the derivative of y in terms of \(f(x), g(x), f^{\prime}(x)\), and \(g^{\prime}(x) .\) \(y=\frac{x^{2}}{f(x)+g(x)}\)

Short Answer

Step by step solution

Apply the Quotient Rule

Differentiate the Numerator

Differentiate the Denominator

Substitute into the Quotient Rule

Simplify the Expression

Key Concepts

Differentiation

Quotient Rule

Derivative

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