Problem 123 - Calculus Volume 1

Question

Assume that \(f(x)\) and \(g(x)\) are both differentiable functions for all \(x\). Find the derivative of each of the functions \(h(x)\). $$ h(x)=x^{3} f(x) $$

Step-by-Step Solution

Answer

The derivative is \(h'(x) = 3x^2 f(x) + x^3 f'(x)\).

Step 1: Applying the Product Rule

To find the derivative of the product of two functions, recall the product rule: if you have functions \(u(x)\) and \(v(x)\), then the derivative of their product \(u(x)v(x)\) is \(u'(x)v(x) + u(x)v'(x)\). In our case, \(u(x) = x^3\) and \(v(x) = f(x)\).

Step 2: Differentiating the Terms

Differentiate \(u(x) = x^3\) to get \(u'(x) = 3x^2\). Since \(f(x)\) is differentiable, denote its derivative by \(f'(x)\).

Step 3: Applying the Product Rule Formula

Substitute \(u(x)\), \(u'(x)\), \(v(x)\), and \(v'(x)\) into the product rule formula: \[ h'(x) = u'(x)v(x) + u(x)v'(x) = 3x^2 f(x) + x^3 f'(x). \]

Step 4: Final Expression for the Derivative

The derivative of \(h(x) = x^3 f(x)\) is given by \(h'(x) = 3x^2 f(x) + x^3 f'(x)\).

Key Concepts

Product Rule

The Product Rule is a fundamental concept in calculus that helps us take the derivative of a product of two functions. This rule is especially useful when dealing with functions that are not easily differentiable individually but are part of a product in another function. For two functions, say \(u(x)\) and \(v(x)\), the product we want to differentiate is \(u(x)v(x)\).

To apply the Product Rule, you need to:

• Find the derivative of the first function \(u(x)\), denote it as \(u'(x)\).
• Find the derivative of the second function \(v(x)\), denote it as \(v'(x)\).
• Apply the formula: \( (uv)' = u'v + uv' \).

This means that you first multiply the derivative of the first function by the second function and add it to the product of the first function and the derivative of the second function. This is useful when our original function is a complex combination of two separate functions.

Derivative of a Function

The derivative of a function is a cornerstone idea in calculus. It measures how a function's output value changes as its input changes. Think of the derivative as the rate of change or the slope of the function at any given point. Mathematically, if you have a function \(f(x)\), the derivative, often written as \(f'(x)\), gives the slope of \(f(x)\) at each point \(x\).

When you derive a function, you are essentially looking to calculate this slope for continuously many points to understand the behavior of the function over its domain:

• The process of finding a derivative is called differentiation.
• Derivatives help evaluate rates of change in applications such as physics, economics, and biology.
• It is symbolically represented by \(d/dx\), where \(x\) indicates what variable we are differentiating with respect to.

Mastering derivatives unlocks deeper understanding of how functions behave and interact.

Differentiable Functions

Differentiable functions are those functions that have a derivative at each point in their domain. In simpler terms, if a function is differentiable, you can draw a tangent line at every point of its graph. This indicates that the function has no sharp corners or discontinuities, and it is smooth throughout its range.

Characteristics of differentiable functions include:

• Smooth curves without any breaks, holes, or cusps.
• Continuous – if a function is differentiable at a point, it is also continuous at that point.
• Allows us to apply differentiation rules, such as the Product Rule, effectively.

Understanding whether a function is differentiable is crucial because it informs us about the types of calculus techniques we can apply to analyze the function further. Differentiable functions integrate well with various calculus concepts, opening doors to exploring tangents, slopes, and the intricate changes in mathematical modeling of real-world phenomena.