Question

Assuming the first and second derivatives of \(f\) and \(g\) exist at \(x,\) find a formula for \(d^{2} / d x^{2}(f(x) g(x)).\)

Short answer

Question: Find the formula for the second derivative of the product of two functions \(f(x)g(x)\). Answer: The second derivative of the product of two functions \(f(x)g(x)\) is given by the formula: $$\frac{d^{2}}{dx^{2}}(f(x)g(x)) = f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)$$

Step-by-step solution

Find the first derivative of \(f(x)g(x)\) using the product rule.

We have \(h(x) = f(x)g(x)\). Let's find the first derivative, \(h'(x)\): $$\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$$

Find the second derivative of \(f(x)g(x)\) using the product rule.

To find the second derivative, \(h''(x)\), we'll differentiate the first derivative expression \(f'(x)g(x) + f(x)g'(x)\) with respect to \(x\). We'll use the product rule again: $$\frac{d^{2}}{dx^{2}}(f(x)g(x)) = \frac{d}{dx}(f'(x)g(x) + f(x)g'(x)) = (f''(x)g(x) + f'(x)g'(x)) + (f'(x)g'(x) + f(x)g''(x))$$

Combine the terms of the second derivative.

Combining the terms in the second derivative expression, we have: $$\frac{d^{2}}{dx^{2}}(f(x)g(x)) = f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x)$$ And this is our final formula for the second derivative of \(f(x)g(x)\).

Key concepts

Product Rule

The product rule is a fundamental principle used in calculus to find the derivative of the product of two functions. Simply put, if you have two functions that are multiplied together, the product rule allows you to differentiate this product. The rule states that the derivative of a product of two functions is given by the derivative of the first function times the second function plus the first function times the derivative of the second function.

For example, given two functions, \( f(x) \) and \( g(x) \), the derivative of their product \( f(x)g(x) \) is computed as:\[ \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) \]

Understanding this rule is crucial for differentiating more complex expressions, especially when dealing with the multiplication of variable functions.

First Derivative

The first derivative of a function symbolizes the rate at which the function's value changes with respect to change in the input value. It provides crucial insights into the behavior of the function, such as identifying points where the function is increasing or decreasing, and finding local maxima and minima. Computationally, the first derivative is the limit of the function's average rate of change as the interval of change approaches zero.

Calculated either from first principles or by using rules of differentiation like the product rule, the first derivative is vital for understanding the fundamental behavior of functions and provides the groundwork for further principles in calculus, such as the second derivative.

Second Derivative

The second derivative is essentially the derivative of the derivative and provides an understanding of the curvature or concavity of a function. It gives information about the acceleration of the function's rate of change. When the second derivative is positive, the function is concave upward, which often indicates a local minimum. Conversely, a negative second derivative implies the function is concave downward and suggests a local maximum.

The process of finding the second derivative can involve applying differentiation rules twice. As seen in the exercise, the product rule was applied to each term of the first derivative to obtain:\[ \frac{d^{2}}{dx^{2}}(f(x)g(x)) = f''(x)g(x) + 2f'(x)g'(x) + f(x)g''(x) \]

Being able to compute and interpret the second derivative is invaluable for thoroughly understanding the nature and shape of graphs and for solving more advanced problems in calculus.

Differentiation

Differentiation is the process of finding the derivative of a function. It is a central operation in calculus that measures how a function changes as its input changes. Differentiation helps to find rates of change, slopes of curves, and many other important concepts that involve change. Differentiation can be performed using several techniques and rules, such as the power rule, the chain rule, the product rule, and the quotient rule.

Differentiation is not just a single step process. It can be expanded to finding higher-order derivatives, such as the second derivative. Mastery of differentiation is essential not only for academic success in mathematics but also for its widespread applications in physics, engineering, economics, and beyond. By understanding differentiation, students can solve complex problems involving motion, optimization, and changes in various systems.