Problem 3 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=\left(1-3 x^{2}\right)^{4}\)

Step-by-Step Solution

Answer

The derivative of the function is \( f'(x) = -24x(1 - 3x^2)^3 \).

Step 1: Identify the Differentation Rule

The function \( f(x) = \left(1 - 3x^2\right)^4 \) is a composite function, which requires using the chain rule to differentiate. The chain rule states if you have a function \( h(x) = g(u(x)) \), then the derivative \( h'(x) = g'(u(x)) \cdot u'(x) \).

Step 2: Differentiate the Outer Function

The outer function here is \( g(u) = u^4 \). Differentiating \( g(u) \) with respect to \( u \) gives \( g'(u) = 4u^3 \).

Step 3: Differentiate the Inner Function

The inner function is \( u(x) = 1 - 3x^2 \). Differentiating \( u(x) \) with respect to \( x \) gives \( u'(x) = -6x \).

Step 4: Apply the Chain Rule

Now, we apply the chain rule: \( f'(x) = g'(u(x)) \cdot u'(x) = 4(1 - 3x^2)^3 \cdot (-6x) \).

Step 5: Simplify the Result

Finally, simplifying the expression gives: \( f'(x) = -24x(1 - 3x^2)^3 \).

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves the calculation of a derivative, which represents how a function changes as its input changes. This process is essential for understanding how a function behaves and is imperative for analyzing rates of change in various applications from physics to finance. The derivative itself is defined as the limit of the ratio of the change in the function's value to the change in the input value, as the change in the input approaches zero.

• The derivative gives us the slope of the tangent line to the function at any point.
• It's used to find local maxima and minima, which are points where a function reaches its highest or lowest value in a small interval.
• In practical terms, it helps in determining instantaneous rates of change, such as speed or growth rates in different contexts.

In our exercise, differentiation is used to find the rate at which the function \(f(x) = (1 - 3x^2)^4\) changes with respect to \(x\). The aim is to calculate its derivative by applying the chain rule, a technique suited for composite functions.

Composite Function

A composite function, as the name suggests, is formed by combining two functions where the output of one function becomes the input of another. Mathematically, it's expressed as \(h(x) = g(u(x))\), where \(g\) and \(u\) are individual functions.

• The exterior function, \(g\), acts on the output of the interior function, \(u\).
• Understanding composite functions is crucial because many real-world functions are naturally composite.

In the exercise given, we see that \(f(x) = (1 - 3x^2)^4\) is a composite function. Here:

• The exterior function is \(g(u) = u^4\), representing a polynomial.
• The interior function is \(u(x) = 1 - 3x^2\), which is a simple quadratic.

The composite structure requires us to differentiate using the chain rule, which effectively handles the layered nature of these functions. This becomes especially handy when the functions are not straightforward to differentiate outright.

Derivative Calculation

The derivative calculation of a composite function, as demonstrated in the exercise, relies on the chain rule—a powerful tool in calculus. The chain rule is formulated to find the derivative of composite functions efficiently. Here’s how it works:

• First, differentiate the exterior function \(g(u) = u^4\) with respect to its input \(u\), yielding \(4u^3\).
• Next, differentiate the interior function \(u(x) = 1 - 3x^2\) with respect to \(x\), resulting in \(-6x\).
• The chain rule then combines these results by multiplying them: \(f'(x) = g'(u(x)) \cdot u'(x) = 4(1 - 3x^2)^3 \cdot (-6x)\).
• This multiplication accounts for the rate of change contributed by both the outer and inner functions.

After applying the formula and multiplying, the derivative, \(f'(x)\), simplifies to \(-24x(1 - 3x^2)^3\). This expression tells us how \(f(x)\) changes with \(x\), capturing the nature of the combined rates of change effectively managed by the chain rule. The chain rule's structured approach provides clarity in dealing with the complex interactions inherent in composite functions.