Tasks: Chain Rule & Implicit Differentiation

Session 05-04 Practice Problems

1 Problem 1: Basic Chain Rule (x)

Differentiate the following functions using the chain rule:

  1. \(f(x) = (3x + 7)^6\)

  2. \(g(x) = \sqrt{5x - 2}\)

  3. \(h(x) = (x^2 - 4x + 1)^{10}\)

  4. \(k(x) = \frac{1}{(2x + 3)^4}\)

2 Problem 2: Chain Rule with Product Rule (xx)

Differentiate the following functions:

  1. \(f(x) = x^3(2x - 1)^4\)

  2. \(g(x) = (x^2 + 1)^2(3x - 5)^3\)

  3. \(h(x) = \frac{x^2}{(x + 1)^3}\)

3 Problem 3: Simplifying Before Differentiating (x)

Differentiate by simplifying first:

  1. \(f(x) = \sqrt{(2x + 1)^3}\)

  2. \(g(x) = \frac{1}{\sqrt{x^2 + 1}}\)

4 Problem 4: Implicit Differentiation - Business Contexts (x)

Find the derivative for each business relationship:

  1. A company’s price \(p\) and quantity \(q\) satisfy a constant revenue constraint: \(pq = 5000\). Find \(\frac{dq}{dp}\).

  2. Marketing spend \(M\) and sales \(S\) follow: \(MS = 12000\). Find \(\frac{dS}{dM}\).

  3. Budget constraint: \(30L + 50K = 9000\) where \(L\) = labor hours and \(K\) = capital units. Find \(\frac{dK}{dL}\).

5 Problem 5: Related Rates - Customer Growth (xx)

A company’s revenue \(R\) (in €1000) depends on its customer base \(C\) (in thousands): \[R = 80\sqrt{C}\]

The company is gaining 2,000 new customers per month.

  1. Find \(\frac{dR}{dC}\) (marginal revenue per customer).

  2. Find \(\frac{dR}{dt}\) in terms of \(C\) and \(\frac{dC}{dt}\).

  3. How fast is revenue growing when \(C = 100\) (100,000 customers)?

  4. How fast is revenue growing when \(C = 400\)?

  5. Explain why revenue growth slows as the customer base grows.

6 Problem 6: Related Rates - Production and Profit (xx)

A retail chain’s annual profit \(P\) (in €1000) and number of stores \(n\) are related by: \[P = 150\sqrt{n} - 4n\]

The company opens 3 new stores per year.

  1. Find \(\frac{dP}{dn}\) and interpret its meaning.

  2. Find \(\frac{dP}{dt}\) when \(n = 25\) stores.

  3. Find \(\frac{dP}{dt}\) when \(n = 100\) stores.

  4. At how many stores does profit stop growing (i.e., \(\frac{dP}{dn} = 0\))?

7 Problem 7: Related Rates - Market Share (xxx)

A company’s profit \(P\) (in €1000) and market share \(m\) (as a percentage) are related by: \[P = 800m - 15m^2\]

Market share is increasing at 1.5% per month.

  1. Find \(\frac{dP}{dt}\) in terms of \(m\).

  2. How fast is profit changing when \(m = 10\%\)?

  3. How fast is profit changing when \(m = 25\%\)?

  4. At what market share is profit maximized?

  5. If the company currently has \(m = 30\%\), should they continue trying to grow market share? Why or why not?