Question

Higher-order derivatives 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 \(y = \log_{10}x\) is \(\frac{-1}{x^2\ln 10}\).

Step-by-step solution

Find the first derivative of the function

The given function is \(\log _{10} x\). To find its first derivative, we can use the formula for the derivative of a logarithmic function: $$\frac{d}{dx}(\log_ax) = \frac{1}{x\ln a}$$ So, the first derivative for our function is: $$\frac{d}{dx}(\log_{10}x) = \frac{1}{x\ln 10}$$

Find the second derivative of the function

Now that we have the first derivative, we can find the second derivative by taking the derivative of the first derivative. We have: $$\frac{d^2}{dx^2}\left(\frac{1}{x\ln 10}\right)$$ To do this, we can use the power rule and rewrite the first derivative as: $$\frac{d^2}{dx^2}\left(x^{-1}\cdot\frac{1}{\ln 10}\right)$$ Now, applying the power rule to find the second derivative: $$\frac{d^2}{dx^2}\left(x^{-1}\cdot\frac{1}{\ln 10}\right) = -1\cdot x^{-2}\cdot\frac{1}{\ln 10}=\frac{-1}{x^2\ln 10}$$

Write the final answer

The second derivative of the given function is: $$\frac{d^2}{dx^2}(\log_{10}x) = \frac{-1}{x^2\ln 10}$$

Key concepts

First Derivative

The first derivative is a foundational concept in calculus, representing the rate of change of a function. In simple terms, it tells us how much the function's output value changes for a small change in the input value. To obtain the first derivative, we use differentiation rules---a set of formulas and techniques.

For our function \(\log_{10} x\), the formula for the derivative of a logarithmic function, such as \(\log_a x\), comes into play:

• The derivative formula is \(\frac{1}{x \ln a}\).

This implies that the derivative of \(\log_{10} x\)
turns into \(\frac{1}{x \ln 10}\).
This result gives us the slope of the tangent line to the \(\log_{10} x\) curve at any point \(\left(x, \log_{10} x\right)\) and is crucial for understanding how \(\log_{10} x\) changes at various points along its curve.

Second Derivative

The second derivative builds upon the first derivative by indicating the rate of change of the rate of change. This helps us understand the curvature or concavity of the original function graph.

• If the second derivative is positive, the original function is concave up.
• If the second derivative is negative, it is concave down.

In finding the second derivative, we start with our first derivative, \(\frac{1}{x \ln 10}\). This expression can be rewritten using negative exponents:

\(x^{-1} \cdot \frac{1}{\ln 10}\).

Differentiating this expression with respect to \(x\) involves using the power rule, which states that the derivative of \(x^n\) is \(n \cdot x^{n-1}\).

Applying this technique, we derive:
\(\frac{d^2}{dx^2} \left( x^{-1} \cdot \frac{1}{\ln 10} \right) = \frac{-1}{x^2 \ln 10}\).
Thus, our second derivative captures how the rate of change of \(\log_{10} x\) itself is changing, shedding light on the function's overall shape.

Logarithmic Functions

Logarithmic functions are mathematical functions defining how numbers are exponentiated to result in given values. Given in the form \(\log_a x\), where \(a\) is the base, logarithmic functions are the inverse of exponential functions.

Understanding logarithms involves grasping key properties:

• The logarithm of a product is the sum of logarithms: \(\log_a (mn) = \log_a m + \log_a n\).
• The logarithm of a quotient is the difference: \(\log_a \left(\frac{m}{n}\right) = \log_a m - \log_a n\).
• The logarithm of a power is scaled by the exponent: \(\log_a (m^n) = n \cdot \log_a m\).

Specific to our calculation, \(\log_{10} x\) uses base 10, commonly known as the common logarithm.
Logarithms simplify multiplicative processes into additive ones, a quality exploited in various mathematical and scientific applications. Calculating derivatives of these functions further aids in understanding changes and trends in periodic phenomena.