Problem 75 - Thomas Calculus

Question

Find the derivative of \(y\) with respect to the given independent variable. \begin{equation}y=x^{3} \log _{10} x\end{equation}

Step-by-Step Solution

Answer

The derivative is \( y' = 3x^2 \log_{10} x + \frac{x^2}{\ln(10)} \).

Step 1: Understand the Problem

We need to find the derivative of the given function \(y = x^3 \log_{10} x\) with respect to the variable \(x\). The function combines polynomial and logarithmic expressions.

Step 2: Product Rule Identification

The function \(y = x^3 \log_{10} x\) is a product of two functions: \(u = x^3\) and \(v = \log_{10} x\). We will use the product rule to find the derivative: \( (uv)' = u'v + uv' \).

Step 3: Differentiate \(u = x^3\)

Find the derivative of \(u = x^3\) with respect to \(x\):\[ u' = \frac{d}{dx} x^3 = 3x^2 \].

Step 4: Differentiate \(v = \log_{10} x\)

Use the formula for the derivative of \(\log_{10} x\):\[ v' = \frac{d}{dx} \log_{10} x = \frac{1}{x \ln(10)} \].

Step 5: Apply Product Rule

Using the product rule \( (uv)' = u'v + uv' \):- Substitute \(u' = 3x^2\), \(v = \log_{10} x\), and \(v' = \frac{1}{x \ln(10)}\) into the product rule: \[ y' = 3x^2 \cdot \log_{10} x + x^3 \cdot \frac{1}{x \ln(10)} \].

Step 6: Simplify the Expression

Simplify the derivative expression:- The second term simplifies to: \[ \frac{x^3}{x \ln(10)} = \frac{x^2}{\ln(10)} \].Combining both terms gives:\[ y' = 3x^2 \cdot \log_{10} x + \frac{x^2}{\ln(10)} \].

Key Concepts

Product Rule

When you deal with the differentiation of a product of two functions, the product rule is your best friend. In calculus, the product rule is a fundamental technique to find the derivative of a function composed of two differentiable functions. It states that if you have a function expressed as the product \(y = uv\), where \(u\) and \(v\) are functions of \(x\), the derivative \(y'\) is given by the formula:

• \((uv)' = u'v + uv'\)

This means you take the derivative of the first function \(u\) and multiply it by the second function \(v\), then add the product of the first function \(u\) and the derivative of the second function \(v'\). Applying this rule correctly is crucial to breaking down complex products into manageable pieces.
In our exercise, we used the product rule to differentiate the function \(y = x^3 \log_{10} x\) by identifying \(u = x^3\) and \(v = \log_{10} x\). This allows us to split the complex problem into simpler sub-steps involving basic derivatives.

Logarithmic Differentiation

Logarithmic differentiation is a technique used when differentiating functions that involve logarithms, especially when combined with other functions. When differentiating a function like \(\log_{b}(x)\), where \(b\) is the base, the derivative is calculated using the formula:

• \(\frac{d}{dx} \log_b x = \frac{1}{x \ln(b)}\)

This result derives from the fact that logarithms convert multiplication into addition and exponentiation into multiplication, making them powerful tools for simplifying products and powers before differentiation.
In the exercise, we used this principle to differentiate \(v = \log_{10} x\), an essential part of applying the product rule. Remembering that \(\ln(10)\) is the natural logarithm of the base 10, we can substitute it into our derivative formula and use it to solve for \(v'\). Securing the correct derivative of logarithmic terms is key when they're part of more complicated expressions.

Calculus

Calculus is the branch of mathematics focusing on limits, functions, derivatives, integrals, and infinite series. It is a powerful tool for solving problems in science and engineering that involve changing quantities. Differentiation, one of calculus's key operations, allows us to determine the instantaneous rate of change or the slope of a function at any given point.
Differentiation is what we did in the exercise to compute the derivative of the function \(y = x^3 \log_{10} x\). By using the product rule and logarithmic differentiation methods, we simplified this complex multiplication of two different nature functions into a solvable differentiation problem.
Calculus helps to break down complex real-world problems involving motion, growth, and change into structured mathematical frameworks that are much easier to work with.

Derivative of Polynomial Function

Polynomials are one of the most common types of functions you'll encounter in calculus. Differentiating a polynomial is relatively straightforward since they are composed of powers of \(x\) with constant coefficients.
To find the derivative for a polynomial function like \(x^n\), use the basic power rule:

• \(\frac{d}{dx} x^n = nx^{n-1}\)

This rule suggests that you multiply the term by its power and reduce the power by one. Since polynomials can be applied to everyday problems like calculating areas, they're significant across mathematics similar to our step-by-step solution here.
In the context of our exercise, the term \(u = x^3\) is a straightforward polynomial, and finding its derivative involves applying this power rule to achieve \(u' = 3x^2\). Once you master these basic derivatives, they serve as building blocks for handling more intricate calculus challenges.