Problem 7 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=\sqrt{3-x^{3}}\)

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Function

First, we should rewrite the square root expression in exponential form to make differentiation easier. The function is given by \( f(x) = \sqrt{3-x^3} \). We can rewrite this as \( f(x) = (3-x^3)^{1/2} \).

Step 2: Apply the Chain Rule

To differentiate \( f(x) = (3-x^3)^{1/2} \), we need to use the chain rule. The chain rule states that if you have a composite function \( y = g(h(x)) \), then the derivative \( y' = g'(h(x)) \cdot h'(x) \).

Step 3: Differentiate the Outer Function

The outer function here is \( g(u) = u^{1/2} \). The derivative of \( u^{1/2} \) with respect to \( u \) is \( \frac{1}{2}u^{-1/2} \).

Step 4: Differentiate the Inner Function

The inner function is \( h(x) = 3-x^3 \). The derivative of \( h(x) \) with respect to \( x \) is \( h'(x) = -3x^2 \).

Step 5: Apply the Chain Rule to Get the Derivative

Using the chain rule, \( f'(x) = \frac{1}{2}(3-x^3)^{-1/2} \cdot (-3x^2) \). This simplifies to \( f'(x) = -\frac{3x^2}{2\sqrt{3-x^3}} \).

Step 6: Simplify the Expression

Multiply the fractions and simplify: The expression is already simplified as: \( f'(x) = -\frac{3x^2}{2\sqrt{3-x^3}} \).

Key Concepts

Chain Rule

When dealing with calculus differentiation, the chain rule is one of the master tools for finding derivatives, especially for composite functions. A composite function is essentially a function within another function. The chain rule helps us differentiate these complex expressions by breaking them down into simpler parts. Let's say you have a function y = f(g(x)), where f is an outer function and g is an inner function. The chain rule formula gives the derivative of y with respect to x as:

\[\frac{dy}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}\]
This means you first differentiate the outer function with respect to the inner function. Then, multiply it by the derivative of the inner function with respect to x. In our original exercise, the function was a composite one, and applying the chain rule helped us efficiently and accurately derive it.

Exponential Form

The exponential form is used to simplify more complex power functions so they can be easily differentiated. In the exercise, the original function was \(f(x) = \sqrt{3-x^3}\). Handling a square root directly can be cumbersome. By converting it into \((3-x^3)^{1/2}\), we employed the power rule for differentiation, which is more straightforward for powers expressed as exponents.

Generally, converting root expressions into exponential form involves using fractional exponents; for example:

• \(\sqrt{x}\) becomes \(x^{1/2}\)
• \(\sqrt[3]{x}\) changes to \(x^{1/3}\)

This approach maximizes the efficiency of differentiation and applies to any expression where roots are present. Such transformations enable the use of powerful calculus tools, like the chain rule, to tackle differentiation problems. For the function in question, transforming into exponential form was the stepping stone that led to its successful differentiation.

Derivative of Composite Functions

Understanding the derivative of composite functions is pivotal in mastering calculus differentiation. Composite functions blend two or more functions, requiring strategic techniques like the chain rule.

When differentiating composite functions, each layer needs separate attention:

• Outer Function: Differentiated first with respect to the inner function.
• Inner Function: Differentiated with respect to the independent variable, like x.

In the exercise problem, our composite function was expressed as \((3-x^3)^{1/2}\), consisting of an outer function \(u^{1/2}\) and an inner function \(3-x^3\). The derivative of the outer function, \(\frac{1}{2}u^{-1/2}\), is multiplied by the derivative of the inner function, \(-3x^2\), according to the chain rule. This results in:

\[f'(x) = \frac{1}{2}(3-x^3)^{-1/2} \cdot (-3x^2) = -\frac{3x^2}{2\sqrt{3-x^3}}\]
By systematically handling each part, we ensure the composite function is fully differentiated, yielding an accurate result. Mastery of differentiating these functions leads to further understanding of more complex calculus problems.