Problem 35 - Calculus an Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative, $$ f(x)=x\left(1-\frac{2}{x+1}\right) $$

Step-by-Step Solution

Answer

The derivative of the given function \(f(x)=x\left(1-\frac{2}{x+1}\right)\) is \(f'(x) = 1 - 2*\frac{1}{(x+1)^2}\). The power rule, product rule and the quotient rule were used to find the derivative.

Step 1: Simplify the Function

First rewrite the expression for easier handling. We can distribute the 'x' inside the parentheses, which will give us: \(f(x) = x - \frac{2x}{(x+1)}.\)

Step 2: Further Simplify the Function

Let's further simplify second term by separating '2x' into '2' and 'x', so we have: \(f(x) = x - 2*\frac{x}{(x+1)}.\)

Step 3: Apply the Derivative Rules

Now we will differentiate the function. The derivative of 'x' is 1 (using power rule). Now, for the second part '-2*\frac{x}{(x+1)}', we use the quotient rule, \( (u/v)' = u'v - uv' / v^2 \), where \( u=x \), \( v=x+1 \), \( u'=1 \) and \( v'=1 \). We also have the constant '2' which remains the same during differentiation. By applying the quotient rule, we get the derivative of the second term as: \((-2*(x)'*(x+1) - 2*x*(x+1)') / (x+1)^2 \), which simplifies to \(-2*\frac{x+1 - x}{(x+1)^2} = -2*\frac{1}{(x+1)^2}.\)

Step 4: Combine the Derivatives

Now, we can state the derivative of the function. The derivative of the two terms combined is: \(f'(x) = 1 - 2*\frac{1}{(x+1)^2}\).

Key Concepts

Power Rule

Understanding the power rule is essential for differentiation in calculus. In essence, if you have a function where the variable, let's say 'x', is raised to a power, such as in \(x^n\), the power rule states that the derivative of this function is \(n*x^{n-1}\). This means you bring the exponent down in front of the variable and subtract one from the exponent.

For example, in our exercise where we have the term \(x\), this can also be seen as \(x^1\). When applying the power rule, the derivative is \(1*x^{1-1}\), which simplifies to \(1*x^0\) or just \(1\) since any number to the zero power is one. The power rule is a swift and effective way to differentiate powers of x, and it's often one of the first rules taught in calculus due to its simplicity and wide application.

Quotient Rule

The quotient rule is a method used when differentiating a function that is the quotient of two other functions. It mathematically expresses how the rate of change (derivative) of a ratio is affected by the rates of change of the numerator and the denominator.

Let's denote the numerator as \(u\) and the denominator as \(v\). The quotient rule states that the derivative of \(u/v\) is \(\frac{u'v - uv'}{v^2}\). Here, \(u'\) and \(v'\) denote the derivatives of \(u\) and \(v\) respectively. In our exercise, we had to differentiate \(\frac{x}{x+1}\), which meant \(u=x\) and \(v=x+1\). The derivatives are \(u'=1\) and \(v'=1\). After applying the quotient rule as per the solution, we got \(\frac{1(x+1) - x(1)}{(x+1)^2}\), which then simplifies to \(\frac{1}{(x+1)^2}\). The quotient rule is especially useful when the function you're differentiating is not easily simplified into a single expression that can be differentiated directly with the power rule.

Differentiation Steps

Solving differentiation problems typically involves a series of logical and methodical steps to streamline the process. These steps help in ensuring that complex functions are tackled efficiently.

The first step is often simplifying the function to make the differentiation process smoother. This can include distributing the variable across terms, combining like terms, or factoring.

The next step is to apply known differentiation rules, such as the power rule or quotient rule, as required by each term of the simplified function. The differentiation step varies depending on the functions involved; however, being systematic can greatly reduce errors.

Lastly, combine the derivatives of all terms to get the full derivative of the original function. In the given exercise, after using the power rule for the first term and the quotient rule for the second, the derivative terms were combined to give the final answer, \(f'(x) = 1 - 2*\frac{1}{(x+1)^2}\). Careful attention during each differentiation step can greatly enhance understanding and accuracy in calculus problems.