Problem 10 - Calculus for Biology and Medicine

Question

Differentiate the functions with respect to the independent variable. \(f(x)=\frac{2}{\left(1-5 x^{2}\right)^{3}}\)

Step-by-Step Solution

Answer

The derivative of \( f(x)=\frac{2}{\left(1-5 x^{2}\right)^{3}} \) with respect to \( x \) is \( 60x(1 - 5x^2)^{-4} \).

Step 1: Recognize the Function Type

The given function is a rational function where the numerator is a constant and the denominator is a function of the variable. This type of function can be differentiated using the chain rule.

Step 2: Identify the Inner Function

Identify the inner function of the composite function: let \( u = 1 - 5x^2 \). Then, the outer function becomes \( f(u) = \frac{2}{u^3} \).

Step 3: Apply the Inner Function Derivative

Differentiate the inner function \( u = 1 - 5x^2 \) with respect to \( x \). The derivative is \( \frac{du}{dx} = -10x \).

Step 4: Differentiate the Outer Function

Differentiate the outer function \( f(u) = \frac{2}{u^3} \) with respect to \( u \). The derivative is \( \frac{df}{du} = -6u^{-4} \).

Step 5: Apply the Chain Rule

Using the chain rule, the derivative of \( f(x) \) with respect to \( x \) is \( \frac{df}{dx} = \frac{df}{du} \times \frac{du}{dx} \). Substitute the derivatives from Steps 3 and 4 to get \( \frac{df}{dx} = -6u^{-4} \times (-10x) \).

Step 6: Substitute Back Inner Function

Substitute \( u = 1 - 5x^2 \) back into the expression, so the derivative becomes \( 60x(1 - 5x^2)^{-4} \).

Key Concepts

Understanding the Chain Rule

The chain rule is a key differentiation technique, especially useful when dealing with composite functions. Composite functions are simply functions within other functions. Here, understanding each part allows us to find the derivative efficiently. The chain rule states that if you have a composite function, say \( f(g(x)) \), then its derivative is the derivative of the outer function with respect to the inner function, multiplied by the derivative of the inner function itself. It's like peeling an onion layer by layer. You differentiate the outer layer first, followed by the inner layer, and multiply these derivatives.
For instance, in our exercise: we have \( f(x) = \frac{2}{(1-5x^2)^3} \). We identified an inner function \( u = 1 - 5x^2 \) and an outer function \( f(u) = \frac{2}{u^3} \).
The process followed:

• Differentiated \( u \) with respect to \( x \), resulting in \( \frac{du}{dx} = -10x \).
• Differentiated \( f(u) \) with respect to \( u \) to get \( \frac{df}{du} = -6u^{-4} \).
• Applied the chain rule by multiplying these results: \( \frac{df}{dx} = \frac{df}{du} \times \frac{du}{dx} \).

This highlights the importance of unpacking functions carefully to apply the chain rule correctly.

Rational Functions in Differentiation

Rational functions are fractions where both the numerator and the denominator are polynomials. In this exercise, the function \( f(x) = \frac{2}{(1-5x^2)^3} \) is a rational function. Here, the numerator is simply a constant and the complexity lies within the denominator. The good news is that understanding rational functions doesn't have to be daunting.
When differentiating rational functions, particularly when you need to consider composite functions, the chain rule often plays a critical role. Get familiar with:

• Identifying if the function fits into this category.
• Recognizing the easier part (constant in the numerator) and focusing on the challenging part (polynomial in the denominator).
• Utilizing the chain rule to tackle the composite part of the function as seen in our exercise.

Embracing these steps will make differentiation more approachable.

Grasping Composite Functions

Composite functions appear when one function fits inside another. Imagine a Russian nesting doll, where opening one reveals another. This is similar to having \( u = 1 - 5x^2 \) sitting inside \( f(u) = \frac{2}{u^3} \). To differentiate these functions, analyze and differentiate each part separately and use the chain rule to bring them together.
Steps to effectively differentiate composite functions:

• Identify the inner function, often the more complex part.
• Determine the outer function that holds the inner function.
• Apply differentiation separately to both components.
• Combine these using the chain rule.

For example, the exercise shows these actions: the inner function \( u = 1 - 5x^2 \) was differentiated as \( \frac{du}{dx} = -10x \), and the outer function \( f(u) \) was differentiated to give \( \frac{df}{du} = -6u^{-4} \). Finally, the chain rule tied it all together for the complete derivative of the composite function. By breaking down the process, understanding composite functions becomes less intimidating and more systematic.