Question

Find the following higher-order derivatives. $$\frac{d^{2}}{d x^{2}}\left(\log _{10} x\right)$$

Short answer

Answer: The second derivative of the function $$f(x) = \log_{10}x$$ is $$f''(x) = -\frac{1}{x^2\ln(10)}$$.

Step-by-step solution

Calculate the first derivative

First, we need to find the derivative of $$f(x) = \log_{10}x$$. To do this, we can use the chain rule. Recall that if $$f(u) = \log_{10}(u)$$, then the derivative of $$f(u)$$ is $$f'(u) = \frac{1}{u\ln(10)}$$. Let's consider $$u = x$$, so we have $$f'(x) = \frac{1}{x\ln(10)}$$. Now, let's find the second derivative.

Calculate the second derivative

Now that we have the first derivative, we can find the second derivative also by using the chain rule. First, let's rewrite the first derivative as a product of two functions: $$f'(x) = \frac{1}{\ln(10)}\cdot\frac{1}{x}$$. We can consider $$g(x) = \frac{1}{x}$$ and note that $$g'(x) = -\frac{1}{x^2}$$. Then, we can rewrite our first derivative as $$f'(x) = \frac{1}{\ln(10)}\cdot g(x)$$. Using the chain rule, we have: $$f''(x) = \frac{1}{\ln(10)}\cdot g'(x) = \frac{1}{\ln(10)}\cdot\left(-\frac{1}{x^2}\right)$$ Thus, the second derivative of $$f(x) = \log_{10}x$$ is: $$f''(x) = \boxed{-\frac{1}{x^2\ln(10)}}$$

Key concepts

Chain Rule

The chain rule is a fundamental principle in calculus, used to calculate the derivative of a composite function. It's a bit like finding the gears in a complex watch—where each small movement is part of a larger chain reaction.

Let's break it down. Imagine you have two functions, one inside the other, like a Russian doll. When differentiating, you need to take into account both the outer function and the inner function. In technical terms, if you have a function \( h(x) \) that is the composition of two functions \( f(g(x)) \) then the derivative of \( h(x) \), denoted \( h'(x) \), is found by multiplying the derivative of the outer function \( f \) at \( g(x) \) by the derivative of the inner function \( g \) at \( x \): \( h'(x) = f'(g(x)) \cdot g'(x) \).

In our case, the chain rule simplifies the process of differentiating \( \log_{10}x \) since you think of the logarithm as the outer function and \( x \) as the inner function. Understanding and applying the chain rule makes it much easier to tackle complex derivatives.

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that may be tough to handle through standard rules. This method is particularly useful when dealing with products, quotients, or powers of functions.

Instead of directly differentiating the original function, you take the natural logarithm of both sides of the equation, which simplifies the expression. It assists in transforming multiplication into addition, division into subtraction, and powers into multiplication.

For instance, if you have a function \( y = f(x) \) that's difficult to differentiate, you might start by taking the natural logarithm of both sides to get \( \ln(y) = \ln(f(x))\). Then differentiate both sides with respect to \( x \), applying the chain rule when necessary. Finally, solve for \( y' \) to get the derivative of the original function. While not used in our specific exercise, understanding its application is crucial when you come across a problem that may benefit from this approach.

First Derivative Calculation

The first derivative of a function indicates the rate of change or slope of the graph at any given point. Calculating the first derivative is often the initial step when analyzing the behavior of functions.

In our exercise, the first derivative of \( f(x) = \log_{10}x \) is found using the fact that \( \frac{d}{dx}\log_{a}x = \frac{1}{x \ln(a)}\), where \( a \) is the base of the logarithm. So the first derivative is \( f'(x) = \frac{1}{x\ln(10)}\). This tells us how rapidly the logarithm function is changing at any point \( x \) and lays the groundwork for finding higher-order derivatives.

Second Derivative Calculation

The second derivative, often notated as \( f''(x) \), provides information about the concavity of the function's graph and the rate of change of the function's slope. If the second derivative is positive, the function's graph is concave upwards; if it's negative, the graph is concave downwards.

In our exercise, the second derivative calculation starts once we have the first derivative. As shown in the solution, you rewrite the first derivative as a product of \( \frac{1}{\ln(10)} \) and \( \frac{1}{x} \) and then use standard differentiation rules. By recognizing \( g(x) = \frac{1}{x} \) and that \( g'(x) = -\frac{1}{x^2} \) not only do we adhere to the chain rule, but we also obtain the second derivative in a clear, step-by-step fashion. The negative sign of the second derivative \( f''(x) = -\frac{1}{x^2\ln(10)} \) indicates that our function's graph is concave downwards for all \( x \) in its domain.