Problem 56 - Calculus - AP Edition

Question

Find the indicated derivative. $$\lambda=\left(\frac{a u+b}{c u+d}\right)^{6} ; \text { find } \frac{d \lambda}{d u} \quad(a, b, c, d \text { constants })$$

Step-by-Step Solution

Answer

The derivative is \( \frac{d\lambda}{du} = 6(ad - bc) \frac{(au + b)^5}{(cu + d)^7} \).

Step 1: Identify the Outer Function

The function \( \lambda \) given is \( \left(\frac{au + b}{cu + d}\right)^6 \). Recognize this as a power function with the outer function being \( x^6 \). To find the derivative, we'll need to use the chain rule.

Step 2: Apply the Power Rule

Using the power rule, differentiate the outer function. If \( f(u) = (g(u))^6 \), then \( f'(u) = 6(g(u))^5 \cdot g'(u) \). Identify \( g(u) = \frac{au + b}{cu + d} \). So, we initially have: \[ \frac{d\lambda}{du} = 6 \left( \frac{au + b}{cu + d} \right)^5 \cdot \frac{d}{du} \left( \frac{au + b}{cu + d} \right). \]

Step 3: Differentiate the Inner Function Using the Quotient Rule

The inner function is \( g(u) = \frac{au + b}{cu + d} \). Use the quotient rule where \( g(u) = \frac{f(u)}{h(u)} \). The derivative is given by: \[ g'(u) = \frac{f'(u)h(u) - f(u)h'(u)}{(h(u))^2}. \] Here, \( f(u) = au + b \) and \( h(u) = cu + d \). Then, \( f'(u) = a \) and \( h'(u) = c \). Apply these in the rule: \[ g'(u) = \frac{a(cu + d) - c(au + b)}{(cu + d)^2}. \]

Step 4: Simplify the Derivative of the Inner Function

Simplify the expression obtained from the quotient rule:\[ g'(u) = \frac{ac(u) + ad - ac(u) - bc}{(cu + d)^2} = \frac{ad - bc}{(cu + d)^2}. \]

Step 5: Combine Results using the Chain Rule

Now substitute \( g'(u) \) back into the derivative: \[ \frac{d\lambda}{du} = 6 \left( \frac{au + b}{cu + d} \right)^5 \cdot \frac{ad - bc}{(cu + d)^2}. \]

Step 6: Finalize the Expression

The derivative \( \frac{d\lambda}{du} \) is expressed as:\[ \frac{d\lambda}{du} = 6(ad - bc) \frac{(au + b)^5}{(cu + d)^7}. \]

Key Concepts

Chain Rule

The chain rule is a fundamental principle in calculus used to differentiate composite functions.
When you have a function nested inside another function, such as \(\lambda = \left( \frac{au+b}{cu+d} \right)^{6}\), the chain rule helps to break down the differentiation process.
This is achieved by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

• Identify the outer function; in this case, it is the power \((x^6)\) applied to the inner function.
• The inner function here is a quotient \(\frac{au+b}{cu+d}\).

So, the chain rule formula for this scenario becomes:\[\frac{d\lambda}{du} = 6\left(\frac{au+b}{cu+d}\right)^{5} \cdot g'(u)\],
where \(g(u)\) is the inner function and \(g'(u)\) is its derivative.

Power Rule

The power rule is handy when dealing with derivatives involving polynomials or power functions.
This rule states that if you have a function of the form: \(f(u) = (g(u))^n\), then its derivative is \(f'(u) = n(g(u))^{n-1} \cdot g'(u)\).
For the function given, \(\lambda = \left(\frac{au+b}{cu+d}\right)^6\), the outer function is a power function, and we apply the power rule by identifying:

• The exponent \(n=6\).
• The expression \(g(u)\) as the inner function.

So the derivative using the power rule is:\[\frac{d\lambda}{du} = 6\left(\frac{au+b}{cu+d}\right)^5 \cdot g'(u)\].
This captures how the function's power changes with respect to its base's derivative.

Quotient Rule

The quotient rule is called upon when you have a fraction-like function where both the numerator and the denominator depend on the variable we are differentiating with respect to.
For the function \(g(u) = \frac{au+b}{cu+d}\), you need the quotient rule to find \(g'(u)\).
This rule says: If \(g(u) = \frac{f(u)}{h(u)}\), then the derivative is:\[g'(u) = \frac{f'(u)h(u) - f(u)h'(u)}{(h(u))^2}\],

• Here, \(f(u) = au+b\) and \(h(u) = cu+d\).
• Calculate \(f'(u)=a\) and \(h'(u)=c\).

Plug the derivatives into the quotient rule:\[g'(u) = \frac{a(cu+d) - c(au+b)}{(cu+d)^2} = \frac{ad-bc}{(cu+d)^2}\].
This provides the rate of change of the inner function, essential for the chain rule application.