91Ó°ÊÓ

When you differentiate a function that is the quotient of two other functions, the Quotient Rule is what you need. The rule states that for a function of the form \( \frac{u(x)}{v(x)} \), the derivative is given by:
\[ \frac{d}{dx} \frac{u(x)}{v(x)} = \frac{u'(x) v(x) - u(x) v'(x)}{[v(x)]^2} \]
Here:

• \( u(x) \) is the numerator and \( v(x) \) is the denominator.

Let's break it down step-by-step:

• First, find the derivatives of \( u(x) \) and \( v(x) \).
• Then, multiply the derivative of the numerator by the denominator.
• Next, subtract the product of the numerator and the derivative of the denominator.
• Finally, square the denominator and place it under the above result.

Applying this to our function \( y = \frac{7x^3}{(4-9x)^5} \):

• We identified \( u(x) = 7x^3 \) and \( v(x) = (4-9x)^5 \).

The Chain Rule simplifies the process of differentiating composite functions. It's needed when you have a function inside another function, like \( v(x) \) in our example:
\[ v(x) = (4-9x)^5 \]. Let's take a closer look at how the chain rule works:

• The general form of the Chain Rule is: if \( y = f(g(x)) \), then \( \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

In this case,

• Set \( g(x) = 4-9x \)
• and \( f(g) = g^5 \). We first find \( f'(g) \) and then multiply by \( g'(x) \).

Step-by-step:

• First, differentiate the outer function: \( f(g) = g^5 \). So, \( f'(g) = 5g^4 \).
• Then, multiply by the derivative of the inner function: \( g(x) = 4-9x \). So, \( g'(x) = -9 \).
• Combine the results to get: \( f'(g(x)) \cdot g'(x) = 5(4-9x)^4 \cdot (-9) \).

The derivative of \( v(x) = (4-9x)^5 \) is hence \( -45(4-9x)^4 \).

The final step in our function differentiation is simplification, which can often be the trickiest part.
Our function after differentiating using the Quotient Rule was:
\[ \frac{d}{dx} \frac{7x^3}{(4-9x)^5} = \frac{21x^2 (4-9x)^5 - 7x^3 [-45(4-9x)^4]}{[(4-9x)^5]^2} \]
We simplified the numerator first:
\[ 21x^2 (4-9x)^5 + 315x^3 (4-9x)^4 \]
Factor out common terms:
\[ 21x^2 (4-9x)^4 [(4-9x) + 15x] \]
This simplifies further:
\[ 21x^2 (4+6x) (4-9x)^4 \]
The denominator \( (4-9x)^{10} \) also simplifies the fraction:
\[ \frac{21x^2 (4+6x)}{(4-9x)^6} \]
Key tips for simplification:

• Factor out common terms in the numerator and denominator.
• Cancel out identical terms where possible to reduce the expression.

With practice, you will find simplifying expressions will become quicker and more intuitive.