Lesson 13 The Chain Rule and L’Hopital’s Rule

13.1 Lecture Content

Video: The Chain Rule

Video: L’Hospital’s Rule

13.2 Lecture Notes

13.2.1 Derivative of a Composition of Functions (Chain Rule)

If a function is defined as a composition \(y = f(g(x))\), then the derivative is given by the chain rule:

\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]

In Leibniz notation: If \(y = f(u)\) and \(u = g(x)\), then:

\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \]

Example: \[ \text{If } y = \sin(x^2), \text{ then } \frac{dy}{dx} = \cos(x^2) \cdot 2x \]

13.2.2 L’Hôpital’s Rule

L’Hôpital’s Rule provides a method to evaluate limits of indeterminate forms \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

13.2.2.1 Statement of the Rule

Let \(f(x)\) and \(g(x)\) be differentiable near \(a\), and suppose:

  • \(\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0\), or
  • \(\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = \infty\)

and

  • \(g'(x) \ne 0\) near \(a\), and
  • \(\lim_{x \to a} \frac{f'(x)}{g'(x)}\) exists (or is \(\pm\infty\)),

then:

\[ \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \]

Note: The rule also applies as \(x \to \infty\) or \(x \to -\infty\).

Example: \[ \lim_{x \to 0} \frac{\sin x}{x} = \frac{0}{0} \Rightarrow \text{Apply L'Hôpital's Rule:} \quad \lim_{x \to 0} \frac{\cos x}{1} = 1 \]

13.3 Examples

13.3.1 The Chain Rule

Video: Solutions

  1. Use the chain rule to find the derivative of \(10 \sqrt{10x^9 + 3x^5}\).

  2. Use the chain rule to find the derivative of \(\left(2(10x^6 - 9x^{4})\right)^{16}\).

  3. If \(f(x) = \cos(4x + 6)\), find \(f'(x)\) and f’(5)$.

  4. Find the derivative of \(-7 \sin^2(-6x^9)\).

  5. Let \(f(x) = 5 \cos(\cos x)\). Find \(f'(x)\)

  6. Let \(f(x) = (7x^2 - 4)^4 (-9x^2 + 4)^{14}\). Find \(f'(x)\).

  7. Let \(g(s) = (4s - 4)^8\). Find \(g'(s)\), \(g'(5)\), \(g''(s)\) and \(g''(5)\).

13.3.2 Derivatives of Logs and Exponentials

Video: Solutions

  1. Let \(f(t) = 7t^4 - 7t + 6e^t\). Find \(f'(t)\).

  2. Find the derivative of \(f(x) = x^5 e^{6x}\). Find \(f'(x)\).

  3. Let \(f(x) = 5x^7 \ln(x) - \frac{4}{3}x^3\). Find \(f'(x)\).

  4. Let \(f(x) = -4x^2 \ln x\). Find \(f'(x)\) and $f’(e^2)

  5. Differentiate \(g(x) = \frac{e^{8x}}{x^2}\).

  6. If \(f(y) = e^{7y} - e^{-7y}\), find \(f'(y)\).

  7. Let \(f(x) = \ln(x^2 + 10x + 29)\). Find \(f'(x)\).

  8. Let \(f(x) = 4 \ln(\sin x)\). Find \(f''(x)\).

13.4 Practice Problems

Practice the techniques discussed in class and in the online videos by solving the following examples.

  1. Differentiate:

    1. \(f(x) = (3x - 1)^{10}\)

    2. \(y = (1 - x)^{-1}\)

    3. \(v(x) = 3^{2x + 1} + e^{3x + 1}\)

  2. Differentiate:

    1. \(f(x) = (x^2 + 1)^5 \ln x\)

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

    3. \(g(x) = \ln(x^2 + 3)\)

  3. Differentiate:

    1. \(h(x) = \ln \left(\frac{9x}{4x - 2}\right)\)

    2. \(h(x) = 2[(x + 1)(x^2 - 1)]^{-1/2}\)

    3. \(h(x) = 3^{x^2 - x}\)

  4. Verify L’Hopital’s Rule can be applied, and then use it to evaluate the limit. Sometimes you may have to apply the rule more than once.

    1. \[\lim_{x \to 1} \frac{x^2 - 2x + 1}{x^2 - x}\]

    2. \[\lim_{x \to -2} \frac{x^3 + 8}{x^2 + 3x + 2}\]

    3. \[\lim_{x \to \infty} \frac{6x^2 - 5x + 1}{3x^2 - 9}\]

    4. \[\lim_{x \to \infty} \frac{x^2}{e^x}\]

  5. (Applied) The average price of a two-bedroom apartment in downtown New York City during the real estate boom from 1994 to 2004 can be approximated by

\[ p(t) = 0.33 e^{0.16 t} \text{ million dollars } \quad (0 \leq t \leq 10), \]

where \(t\) is time in years (\(t = 0\) represents 1994). What was the average price of a two-bedroom apartment in downtown New York City in 2003, and how fast was it increasing?

  1. Find the equation of the tangent line to \[y = \ln \sqrt{2x^2 + 1}\] at \(x=1\).

13.5 Self Assessment

Time yourself and try to solve the following questions within twenty minutes.

  1. Differentiate: \[r(x) = \left(\sqrt{x+1} + \sqrt{x}\right)^3\]

  2. Differentiate: \[r(x) = \left(e^{2x^2}\right)^3\]

  3. Determine the limit: \[\lim_{x \to -\infty} \frac{4x^3 + x^2 - 2x}{x^2 - 7}\]

  4. Determine the limit: \[\lim_{x \to \frac{1}{2}} \frac{\sin(2x - 1)}{2x - 1}\]

  5. The total spent on research and development by the federal government in the United States during 1995–2007 can be approximated by

\[ S(t) = 7.4 \ln t + 3 \text{ billion dollars} \quad (5 \leq t \leq 19) \]

where \(t\) is the year since 1990. What was the total spent in 2005 \((t = 15)\) and how fast was it increasing?

13.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill D CON COM
Find the derivative using the chain rule.
Apply L’Hopital’s Rule to an indeterminate limit.
Solve applied problems using advanced derivative rules.