Problem 7 - Calculus for Biology and Medicine

Question

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\sqrt{1-x^{3}} $$

Step-by-Step Solution

Answer

The derivative of \( f(x) = \sqrt{1-x^{3}} \) is \( -\frac{3x^2}{2\sqrt{1-x^3}} \).

Step 1: Identify the function

First, we recognize the function we need to differentiate: \( f(x) = \sqrt{1 - x^3} \). This can be rewritten using exponent notation as \( f(x) = (1 - x^3)^{1/2} \).

Step 2: Apply the Chain Rule

The function is a composition of two functions: the outer function is \( u^{1/2} \) and the inner function is \( u = 1 - x^3 \). To differentiate, we will use the chain rule: \( \frac{d}{dx} [f(u)] = f'(u) \cdot \frac{du}{dx} \).

Step 3: Differentiate the outer function

Differentiate the outer function \( u^{1/2} \) with respect to \( u \): \( \frac{d}{du}(u^{1/2}) = \frac{1}{2}u^{-1/2} \).

Step 4: Differentiate the inner function

Differentiate the inner function \( u = 1 - x^3 \) with respect to \( x \): \( \frac{du}{dx} = -3x^2 \).

Step 5: Combine results using the chain rule

Apply the chain rule: \( \frac{d}{dx} (1 - x^3)^{1/2} = \frac{1}{2}(1 - x^3)^{-1/2} \cdot (-3x^2) \).

Step 6: Simplify the derivative

Simplify the expression: \( \frac{d}{dx} f(x) = -\frac{3x^2}{2\sqrt{1 - x^3}} \).

Key Concepts

Chain Rule in Differentiation

The chain rule is a fundamental concept in calculus that helps us differentiate composite functions. It allows us to take the derivative of a function that is made up of two or more simpler functions. The basic idea is to first identify the inner and outer functions in the composite structure.
For instance, consider a function composed of an inner function \( u(x) \) and an outer function \( v(u) \). To find the derivative of the composite function \( v(u(x)) \) with respect to \( x \), the chain rule provides a straightforward approach:

• Dissolve the composite function into its inner and outer components.
• Differentiate the outer function with respect to the inner function \( u \).
• Differentiate the inner function \( u \) with respect to \( x \).
• Multiply these derivatives together to find the overall derivative.

In formula terms, if \( v = f(u) \) and \( u = g(x) \), then the chain rule can be represented as \( \frac{dv}{dx} = \frac{dv}{du} \times \frac{du}{dx} \). This ensures we account for how changes in \( x \) affect \( u \), and subsequently \( v \).
When applying it to a function like \( f(x) = \sqrt{1 - x^3} \), we see how important the chain rule is in obtaining the derivative effectively.

Understanding Derivatives

Derivatives represent a fundamental component of calculus. They measure how a function changes as its input changes. In simple terms, the derivative of a function at a given point tells us the slope of the tangent to the curve at that point.
To calculate a derivative, we often start by recognizing the function's form and applying the relevant rules. These can include the power rule, product rule, quotient rule, or, as seen in many composite functions, the chain rule.
A derivative is not only a measure of rate of change but also a powerful tool for understanding the behavior of functions. It can tell us

• where functions increase or decrease,
• identify the local maxima or minima,
• and provide insight into the concavity of the function.

When differentiating \( f(x) = \sqrt{1-x^3} \), applying the chain rule helps understand how the combination of \( (1-x^3) \) and the square root function changes with \( x \). It's this blend of basic derivative principles and advanced rules that gives us a precise way to analyze the function's behavior.

Composite Functions and Their Role

Composite functions are a type of function where one function is applied to the result of another function. In a composite function \( f(g(x)) \), there's an inner function \( g(x) \) and an outer function \( f(u) \), where \( u = g(x) \). This layered structure can complicate differentiation, making the chain rule essential.
To better understand composite functions, consider an analogy: think of them as a series of machines where each machine takes the output of the previous one as its input. The idea of first running \( g(x) \) and then feeding its output into \( f(u) \) perfectly illustrates this.
When dealing with composite functions, it's crucial to:

• Identify both the outer and inner functions correctly.
• Apply the chain rule consistently to differentiate them.
• Simplify results carefully to avoid errors.

By dissecting the problem of differentiating \( \sqrt{1 - x^3} \), we see the clear separation in task: handle the inner polynomial function separately from the radical outer function. This layered approach not only aids in precision but also deepens our understanding of function dynamics.