Question

In Exercises, find the second derivative of the function. $$ f(x)=3+2 \ln x $$

Short answer

The second derivative of the function \(f(x)=3+2 \ln x\) is \(f''(x) = -2/x^2\)

Step-by-step solution

Find the first derivative

Apply the rule that the derivative of a constant is zero and derivative of \(\ln x\) is \(1/x\) to get the first derivative. \(f'(x) = 0 + 2/x = 2/x\)

Find the second derivative

Apply the rule for the derivative of \(1/x\), which is \(-1/x^2\) to get the second derivative. \(f''(x) = -2/x^2\)

Key concepts

First Derivative

The first derivative of a function is a foundational concept in calculus that represents the rate at which the function's output changes with respect to its input. In the context of our sample exercise, the function given is \(f(x) = 3 + 2 \ln(x)\). When we calculate the first derivative, \(f'(x)\), we're looking for how quickly \(f(x)\) is changing at any point ‘x’.

To find the first derivative of our function, we can use basic differentiation rules. The derivative of the constant, which is ‘3’ in our case, is zero. For the natural logarithm, \(\ln(x)\), the derivative rule is that it equals \(1/x\). Therefore, the first derivative of our function is \(\frac{2}{x}\). This means that for every unit increase in 'x', the function \(f(x)\) increases by twice the reciprocal of 'x'.

Understanding the first derivative is crucial as it sets the foundation for further calculus work, such as finding the second derivative, or analyzing the graph of the function for slopes and tangents.

Natural Logarithm Differentiation

The natural logarithm, denoted as \(\ln(x)\), is an important function in calculus that describes the time needed to reach a certain level of continuous growth. Differentiating natural logarithm functions is a standard procedure in calculus. For the function \(\ln(x)\), the derivative is \(\frac{1}{x}\).

This relationship is significant because it shows the inverse nature of the exponential function and the logarithmic function; while the exponential function grows rapidly, the logarithmic function grows much more slowly. In the exercise provided, we use the rule for differentiating \(\ln(x)\), which comes in handy especially when dealing with growth and decay problems in real-world applications or when analyzing changes in phenomena that can be modeled logarithmically.

Remember, the argument of the natural logarithm must be positive, since \(\ln(x)\) is undefined for zero and negative values. Misapplication of the differentiation rule for \(\ln(x)\) can lead to errors, so one must ensure that the 'x' in \(\ln(x)\) is always within its domain of \(x > 0\).

Calculus Exercises

Calculus exercises, like the ones involving the calculation of second derivatives, are designed to reinforce understanding of the calculus concepts and rules. These exercises often range from simple to complex, involving various functions and requiring the application of different differentiation techniques.

In the exercise under discussion, we progressed from finding the first derivative to calculating the second derivative, \(f''(x)\), with the function \(f(x) = 3 + 2 \ln(x)\). The calculated first derivative, \(f'(x) = \frac{2}{x}\), then becomes the subject of differentiation to obtain the second derivative.

For the second derivative, we apply the rule derivative of \(\frac{1}{x}\), which is \(-\frac{1}{x^2}\), thereby getting \(f''(x) = -\frac{2}{x^2}\). These kinds of exercises are crucial in teaching students not just to perform rote calculations, but to understand the implications of these derivatives in the context of the problem being solved – such as concavity, inflection points, and acceleration in physical applications.