Question

In Exercises \(33-36,\) find the first four derivatives of the function. $$y=\frac{x+1}{x}$$

Short answer

The first four derivatives of the function \(y=(x+1)/x\) are: \(y' = 1/x^2\), \(y'' = -2/x^3\), \(y''' = 6/x^4\), \(y'''' = -24/x^5\).

Step-by-step solution

Differentiate the Function - First time

Using the quotient rule, the first derivative, \(y'\), would be obtained as: \((x+1)'*x - (x+1)*x' / x^2 = (1*x - (x+1)*1) / x^2 = (x + 1 - x) / x^2 = 1 / x^2\)

Differentiate the Function - Second time

The second derivative of the function, \(y''\), is also obtained using the quotient rule: \((y')' = (1)'*x^2 - (1*x^2)' / (x^2)^2 = (0 - 2*x) / (x^4) = -2 / x^3\)

Differentiate the Function - Third time

For finding the third derivative, differentiate the second derivative using the power rule: \((y'')' = (-2 / x^3)' = -2*(-3/x^4) = 6 / x^4\)

Differentiate the Function - Fourth time

Finally, differentiate \(y'''\) using the power rule to find the fourth derivative: \((y''')' = (6 / x^4)' = 6*(-4/x^5) = -24 / x^5\)

Key concepts

Quotient Rule

When differentiating functions in calculus, especially when dealing with ratios of functions, the quotient rule is an essential tool. It allows us to find the derivative of a division between two functions. For example, if we have a function presented as \(y = \frac{u(x)}{v(x)}\), where both \(u\) and \(v\) are functions of \(x\), the quotient rule states that the derivative of \(y\) with respect to \(x\) is: \[y' = \frac{u'v - uv'}{v^2}\] where \(u'\) and \(v'\) are derivatives of \(u\) and \(v\), respectively.

In our exercise, to find the first derivative of \(y = \frac{x+1}{x}\), we label the numerator as \(u(x) = x + 1\) and the denominator as \(v(x) = x\). By applying the quotient rule, the first derivative \(y'\) simplifies to \(\frac{1}{x^2}\). Clearly understanding and correctly applying the quotient rule is crucial for accurately solving calculus exercises involving the division of functions.

Derivative

The derivative is a fundamental concept in calculus representing an instantaneous rate of change or the slope of the tangent line to the graph of a function at any given point. It is the key to solving many problems in various fields including physics, engineering, and economics. Derivatives can be simple, such as when we find the derivative of a constant or a linear function, or they can involve more intricate rules like the power and quotient rules if the function is more complex.

To excel in calculus exercises, it is crucial to understand which derivative rules apply in different scenarios. As with our exercise, identifying that the quotient rule must be used for \(y = \frac{x+1}{x}\) is the first step. Subsequent derivatives require recognizing when to switch to other rules, like the power rule, which is used to find higher-order derivatives in later steps of the exercise.

Power Rule

The power rule is a streamlined technique for differentiating functions that are polynomials or have powers of \(x\). For any function of the form \(f(x) = x^n\), where \(n\) is any real number, the power rule states that its derivative, \(f'(x)\), is \(nx^{n-1}\). This crucial rule simplifies many calculus exercises that involve finding derivatives.

In our given example, after finding the first and second derivatives using the quotient rule, we apply the power rule to the subsequent derivatives. For instance, to find the third derivative of \(y = -2 / x^3\), we apply the power rule to simplify it to \(6/x^4\), avoiding the more complex quotient rule which is not needed here. Understanding when to apply the power rule can greatly streamline the process of differentiation.

Calculus Exercises

Calculus exercises often involve practicing the application of derivative rules to calculate the derivatives of various functions. These exercises help build conceptual understanding and provide proficiency in differentiation techniques. It is vital to first understand the form of the function in question and determine which rule is most appropriate to use – be it the quotient rule, power rule, product rule, or chain rule.

Through the step-by-step solution of the exercise, where we found the first four derivatives of \(y = \frac{x+1}{x}\), we recognize the importance of following a systematic approach and the necessity to switch between different rules as needed. Exercise improvement advice suggests explaining the selection process for choosing the appropriate rule for each step, providing students with the insight to tackle diverse problems they may encounter in calculus.