Problem 55 - Calculus an Applied Approach

Question

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ f(x)=\frac{1}{\left(x^{2}-3 x\right)^{2}} $$

Step-by-Step Solution

Answer

The derivative of the function \(f(x)=\frac{1}{(x^{2}-3 x)^{2}}\) is \(f'(x) = -2 (x^{2} - 3x)^{-1} * (2x - 3)\)

Step 1: Identify the function type

First, acknowledge that this is a composite function in the form of \( g(f(x)) \) = \( \frac{1}{u^{2}} \) where u = \(x^{2} - 3x\). Because \(f(x)\) is in the form of fraction the quotient rule will be needed.

Step 2: Apply the quotient rule

Differentiating with respect to x, we get \(-2 (x^{2} - 3x)^{-1}\)using the power rule. But, we are not done, we have only found the derivative of the outer function.

Step 3: Apply the chain rule

The chain rule states that the derivative of a composition of functions is the product of the derivatives. In this case, that implies that we differentiate the \(u = x^{2} -3x\) which yields \(2x -3\). Therefore, the overall derivative will be \(-2 (x^{2} - 3x)^{-1} * (2x - 3)\).

Key Concepts

Quotient Rule

When finding derivatives of functions that are fractions, the quotient rule is your best friend. This rule helps differentiate a function that is divided by another function. The notation often used for the quotient rule is:

• If you have a function: \( f(x) = \frac{u(x)}{v(x)} \),
• Then the derivative is: \( f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2} \).

In simpler terms, you take:

• The derivative of the numerator \( u'(x) \), and multiply it by the denominator \( v(x) \).
• Then subtract the numerator \( u(x) \), multiplied by the derivative of the denominator \( v'(x) \).
• Finally, place it all over the square of the original denominator.

For the given exercise, though we started with a fraction, simplifying it using the chain rule was more efficient.

Chain Rule

Encountering a composite function? The chain rule is here to help. It's a must-have for differentiating functions nested within each other, resembling layers of an onion. For a function \( h(x) = g(f(x)) \), the chain rule tells us:

• Differentiate the outer layer \( g' \).
• Multiply it by the derivative of the inner layer \( f'(x) \).

Let's look at our exercise:

• Our outer function is \( g(u) = \frac{1}{u^2} \), leading to \( -2u^{-3} \).
• The inner function is \( u = x^2 - 3x \), differentiating to \( 2x - 3 \).

The completed derivative is the multiplication of these two, achieving the link through the layers.

Power Rule in Differentiation

The power rule is one of the simplest yet most powerful tools in calculus. It allows for quick differentiation when dealing with powers of a variable. If a function is \( f(x) = x^n \), then its derivative is:

• \( f'(x) = n \cdot x^{n-1} \).

How does it apply to our problem?

• We transformed the function \( \frac{1}{(x^2 - 3x)^2} \) into \( (x^2 - 3x)^{-2} \) using algebra.
• Applying the power rule, the outer part becomes \( -2 \cdot (x^2 - 3x)^{-3} \).

Whenever a power is involved, the power rule simplifies the differentiation process immensely.